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Theorem ax8dfeq 30948
 Description: A version of ax-8 1979 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 2238 . 2 𝑧 𝑧 = 𝑤
2 ax8 1983 . . . 4 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
32equcoms 1934 . . 3 (𝑧 = 𝑤 → (𝑤𝑥𝑧𝑥))
4 ax8 1983 . . 3 (𝑧 = 𝑤 → (𝑧𝑦𝑤𝑦))
53, 4imim12d 79 . 2 (𝑧 = 𝑤 → ((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦)))
61, 5eximii 1754 1 𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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