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Theorem axextdfeq 30819
Description: A version of ax-ext 2494 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 9166 . . 3 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 ax8 1944 . . . 4 (𝑥 = 𝑦 → (𝑥𝑤𝑦𝑤))
32imim2i 16 . . 3 (((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) → ((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)))
41, 3eximii 1742 . 2 𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤))
5 biimpexp 30725 . . 3 (((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
65exbii 1752 . 2 (∃𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ∃𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
74, 6mpbi 218 1 𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-cleq 2507  df-clel 2510  df-nfc 2644
This theorem is referenced by: (None)
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