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Theorem axexte 2771
 Description: The axiom of extensionality (ax-ext 2770) restated so that it postulates the existence of a set 𝑧 given two arbitrary sets 𝑥 and 𝑦. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
Assertion
Ref Expression
axexte 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axexte
StepHypRef Expression
1 ax-ext 2770 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 19.36v 1994 . 2 (∃𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
31, 2mpbir 234 1 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by: (None)
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