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Theorem cbvalvw 1919
Description: Change bound variable. See cbvalv 1917 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1448. (Revised by Gino Giotto, 25-Aug-2024.)
Hypothesis
Ref Expression
cbvalvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalvw (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvalvw
StepHypRef Expression
1 cbvalvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21spv 1860 . . 3 (∀𝑥𝜑𝜓)
32alrimiv 1874 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
41equcoms 1708 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
54biimprd 158 . . . 4 (𝑦 = 𝑥 → (𝜓𝜑))
65spimv 1811 . . 3 (∀𝑦𝜓𝜑)
76alrimiv 1874 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
83, 7impbii 126 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  cbvralvw  2707  cbvreuvw  2709
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