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Theorem cbvaev 2122
Description: Change bound variable in an equality with a dv condition. Instance of aev 2126. (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 2090 . . 3 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
21cbvalivw 2081 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑡 𝑡 = 𝑦)
3 ax7 2090 . . 3 (𝑡 = 𝑧 → (𝑡 = 𝑦𝑧 = 𝑦))
43cbvalivw 2081 . 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 1622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1846
This theorem is referenced by:  aevlem0  2123  aevlem  2124
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