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Theorem cbvrabv2 39779
Description: A more general version of cbvrabv 3327. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
cbvrabv2.1 (𝑥 = 𝑦𝐴 = 𝐵)
cbvrabv2.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrabv2 {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrabv2
StepHypRef Expression
1 nfcv 2890 . 2 𝑦𝐴
2 nfcv 2890 . 2 𝑥𝐵
3 nfv 1980 . 2 𝑦𝜑
4 nfv 1980 . 2 𝑥𝜓
5 cbvrabv2.1 . 2 (𝑥 = 𝑦𝐴 = 𝐵)
6 cbvrabv2.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
71, 2, 3, 4, 5, 6cbvrabcsf 3697 1 {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1620  {crab 3042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-sbc 3565  df-csb 3663
This theorem is referenced by:  smfsuplem2  41493
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