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Theorem cbvralv2 3836
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvralv2.1 (𝑥 = 𝑦 → (𝜓𝜒))
cbvralv2.2 (𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
cbvralv2 (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒)
Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvralv2
StepHypRef Expression
1 nfcv 2899 . 2 𝑦𝐴
2 nfcv 2899 . 2 𝑥𝐵
3 nfv 1921 . 2 𝑦𝜓
4 nfv 1921 . 2 𝑥𝜒
5 cbvralv2.2 . 2 (𝑥 = 𝑦𝐴 = 𝐵)
6 cbvralv2.1 . 2 (𝑥 = 𝑦 → (𝜓𝜒))
71, 2, 3, 4, 5, 6cbvralcsf 3832 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1542  wral 3053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2372  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-sbc 3681  df-csb 3791
This theorem is referenced by: (None)
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