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Theorem axextdfeq 32033
Description: A version of ax-ext 2750 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 9614 . . 3 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 ax8 2150 . . . 4 (𝑥 = 𝑦 → (𝑥𝑤𝑦𝑤))
32imim2i 16 . . 3 (((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) → ((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)))
41, 3eximii 1911 . 2 𝑧((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤))
5 biimpexp 31929 . . 3 (((𝑧𝑥𝑧𝑦) → (𝑥𝑤𝑦𝑤)) ↔ ((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤))))
65exbii 1923 . 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 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-cleq 2763  df-clel 2766  df-nfc 2901
This theorem is referenced by: (None)
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