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Theorem axextdfeq 31827
 Description: A version of ax-ext 2631 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
axextdfeq 𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))

Proof of Theorem axextdfeq
StepHypRef Expression
1 axextnd 9451 . . 3 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 ax8 2036 . . . 4 (𝑥 = 𝑦 → (𝑥𝑤𝑦𝑤))
32imim2i 16 . . 3 (((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) → ((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)))
41, 3eximii 1804 . 2 𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤))
5 biimpexp 31723 . . 3 (((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
65exbii 1814 . 2 (∃𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ∃𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
74, 6mpbi 220 1 𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-cleq 2644  df-clel 2647  df-nfc 2782 This theorem is referenced by: (None)
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