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Theorem cbvrabv2 44524
Description: A more general version of cbvrabv 3441. Usage of this theorem is discouraged because it depends on ax-13 2366. Use of cbvrabv2w 44525 is preferred. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (New usage is discouraged.)
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 2899 . 2 𝑦𝐴
2 nfcv 2899 . 2 𝑥𝐵
3 nfv 1909 . 2 𝑦𝜑
4 nfv 1909 . 2 𝑥𝜓
5 cbvrabv2.1 . 2 (𝑥 = 𝑦𝐴 = 𝐵)
6 cbvrabv2.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
71, 2, 3, 4, 5, 6cbvrabcsf 3942 1 {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  {crab 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2366  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3431  df-sbc 3779  df-csb 3895
This theorem is referenced by: (None)
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