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Theorem cbvaev 1976
 Description: Change bound variable in an equality with a dv condition. Instance of aev 1980. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
Assertion
Ref Expression
cbvaev (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

Proof of Theorem cbvaev
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 ax7 1940 . . 3 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
21cbvalivw 1931 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑡 𝑡 = 𝑦)
3 ax7 1940 . . 3 (𝑡 = 𝑧 → (𝑡 = 𝑦𝑧 = 𝑦))
43cbvalivw 1931 . 2 (∀𝑡 𝑡 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
52, 4syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  aevlem0  1977  aevlem  1978  axc11nlemOLD2  1985  aevlemOLD  1986  axc11nlemOLD  2157  axc11nlemALT  2305  aevlemALTOLD  2319
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