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Theorem cbvrexvw 2731
Description: Version of cbvrexv 2727 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
Hypothesis
Ref Expression
cbvralvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexvw (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvrexvw
StepHypRef Expression
1 eleq1w 2254 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvralvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2anbi12d 473 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
43cbvexvw 1932 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐴𝜓))
5 df-rex 2478 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
6 df-rex 2478 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
74, 5, 63bitr4i 212 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wex 1503  wcel 2164  wrex 2473
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-clel 2189  df-rex 2478
This theorem is referenced by:  cbvrex2vw  2738  prodmodclem2  11720  prodmodc  11721  zsupssdc  12091  pceu  12433  4sqlem12  12540  nninfdclemcl  12605  grprida  12970  dfgrp2  13099  dfgrp3mlem  13170  lss1d  13879  bj-charfunbi  15303
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