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Destroying a card is not the same as negating its activation or effect. For example, when a player activates "Dark Hole" and the other player Chains "Mystical Space Typhoon" targeting "Dark Hole", it will destroy "Dark Hole", but "Dark Hole" will still resolve, so the monsters are still destroyed.
Such examples of negation are:
If the activation of a Spell or Trap Card is negated, that card is sent to the Graveyard after the entire Chain it was activated in resolves if it has not already been moved somewhere, but it is not treated as being sent from the field.
If the activation of effect that can only be activated a limited number of times is negated, that activation still counts towards the activation limit. For example, if "Chaos Sorcerer's" effect is negated by "Light and Darkness Dragon", it cannot activate its effect again that turn.
To negate an effect is to prevent it from being applied.
Effects that start a Chain resolve for no effect when activated if negated, but can still be activated. For example, even if the effect of "Cannon Soldier" is negated by "Effect Veiler", it can still be activated by Tributing a monster, but the opponent will not take any damage. Unlike negating the activation, a card whose effect has been negated will still resolve (but with no effect); as such, because "Starlight Road" only negates card effects and not activations, any card or effect that would respond to the Summon of "Stardust Dragon" misses the timing, as the card or effect that "Starlight Road" negated resolves after it for no effect.
Continuous Effects are simply not applied.
To negate a Summon is to prevent a monster from being successfully inherently Summoned. A Summon can only be negated during the Summon negation window. A negated Normal Summon still counts as the one Normal Summon/Set per turn.
Only inherent Summons can be negated (Normal Summons, Flip Summons, Synchro Summons, Xyz Summons, Pendulum Summons and Special Summons with a Special Summoning condition), and only when they occur outside of a Chain. For example, the Normal Summon of a monster with "Ultimate Offering" cannot be negated if it was Chained to "Pot of Greed" (as the monster is Summoned in the middle of a Chain), but it can be negated if "Ultimate Offering" is activated as Chain Link 1 (as the monster is Summoned immediately after the effect of "Ultimate Offering" resolves).
If the Summon of a monster(s) is negated, that monster(s) are not considered to have been on the field at the time its Summon was negated (even if it is being Flip Summoned or is a Gemini monster being Normal Summoned again). For example, if a monster's Summon is negated by "Forced Back" while "Degenerate Circuit" is active, that monster is returned to its owner's hand and not banished, because it was not on the field. If a Summon is negated, the monster was not successfully Summoned, so Special Summon-only monsters cannot be Special Summoned by effects other than their specified method.
To negate an attack is to stop a battle from taking place with a card effect after an attack target has been selected or after an attack has been declared. An attack can only be negated during the Battle Step, and if it is, the Damage Step does not occur. Cards which can do this include "Negate Attack", "Hero Barrier", and "Maiden with Eyes of Blue". It is worth noting that many Trap Cards which negate an attack target the monster that declared the attack.
Cards such as "Mirror Force" and "Dimensional Prison" do not negate attacks; instead, the attacking monster leaves the field during the Battle Step and the attack ends; if the monster does not leave the field, the attack continues. For example, if "Total Defense Shogun" attacks while in Defense Position and the opponent activates "Mirror Force", the attack continues as "Total Defense Shogun" is still on the field.