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Theorem axextdfeq 33069
Description: A version of ax-ext 2796 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 10007 . . 3 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 ax8 2121 . . . 4 (𝑥 = 𝑦 → (𝑥𝑤𝑦𝑤))
32imim2i 16 . . 3 (((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) → ((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)))
41, 3eximii 1838 . 2 𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤))
5 biimpexp 32973 . . 3 (((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
65exbii 1849 . 2 (∃𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ∃𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
74, 6mpbi 233 1 𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-clel 2896  df-nfc 2964
This theorem is referenced by: (None)
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