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Theorem axext2 2590
Description: The Axiom of Extensionality (ax-ext 2589) 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 2589 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 19.36v 1890 . 2 (∃𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
31, 2mpbir 219 1 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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