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Theorem cbvralw 2709
Description: Rule used to change bound variables, using implicit substitution. Version of cbvral 2711 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1517 and ax-bndl 1519 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvralw.1 𝑦𝜑
cbvralw.2 𝑥𝜓
cbvralw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvralw (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvralw
StepHypRef Expression
1 nfcv 2329 . 2 𝑥𝐴
2 nfcv 2329 . 2 𝑦𝐴
3 cbvralw.1 . 2 𝑦𝜑
4 cbvralw.2 . 2 𝑥𝜓
5 cbvralw.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvralfw 2705 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1470  wral 2465
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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470
This theorem is referenced by:  pcmptdvds  12356  lgseisenlem2  14691
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