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Theorem cbvralv2 3066
 Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
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 2281 . 2 𝑦𝐴
2 nfcv 2281 . 2 𝑥𝐵
3 nfv 1508 . 2 𝑦𝜓
4 nfv 1508 . 2 𝑥𝜒
5 cbvralv2.2 . 2 (𝑥 = 𝑦𝐴 = 𝐵)
6 cbvralv2.1 . 2 (𝑥 = 𝑦 → (𝜓𝜒))
71, 2, 3, 4, 5, 6cbvralcsf 3062 1 (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∀wral 2416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-sbc 2910  df-csb 3004 This theorem is referenced by: (None)
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