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Theorem axextdfeq 35302
Description: A version of ax-ext 2697 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 10588 . . 3 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 ax8 2104 . . . 4 (𝑥 = 𝑦 → (𝑥𝑤𝑦𝑤))
32imim2i 16 . . 3 (((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) → ((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)))
41, 3eximii 1831 . 2 𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤))
5 biimpexp 35220 . . 3 (((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
65exbii 1842 . 2 (∃𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ∃𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
74, 6mpbi 229 1 𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-clel 2804  df-nfc 2879
This theorem is referenced by: (None)
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