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Theorem axext2 2752
Description: The Axiom of Extensionality (ax-ext 2751) 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
axext2 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axext2
StepHypRef Expression
1 ax-ext 2751 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 19.36v 2072 . 2 (∃𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
31, 2mpbir 221 1 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by: (None)
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