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Theorem ax8dfeq 31678
Description: A version of ax-8 1990 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
Assertion
Ref Expression
ax8dfeq 𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))

Proof of Theorem ax8dfeq
StepHypRef Expression
1 ax6e 2248 . 2 𝑧 𝑧 = 𝑤
2 ax8 1994 . . . 4 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
32equcoms 1945 . . 3 (𝑧 = 𝑤 → (𝑤𝑥𝑧𝑥))
4 ax8 1994 . . 3 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
53, 4imim12d 81 . 2 (𝑧 = 𝑤 → ((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦)))
61, 5eximii 1762 1 𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by: (None)
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