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Theorem cbvralvw 2707
Description: Version of cbvralv 2703 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
Hypothesis
Ref Expression
cbvralvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvralvw (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvralvw
StepHypRef Expression
1 eleq1w 2238 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvralvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2imbi12d 234 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
43cbvalvw 1919 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
5 df-ral 2460 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 df-ral 2460 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
74, 5, 63bitr4i 212 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  wcel 2148  wral 2455
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  df-clel 2173  df-ral 2460
This theorem is referenced by:  cbvral2vw  2714  cc1  7242  zsupssdc  11925  prmpwdvds  12323  grprinvlem  12683  grprinvd  12684  2sqlem6  14089  2sqlem10  14094  bj-charfunbi  14185
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