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Theorem cbvrexv2 2970
Description: Rule used to change the bound variable in a restricted existential 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
cbvrexv2 (∃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 cbvrexv2
StepHypRef Expression
1 nfcv 2220 . 2 𝑦𝐴
2 nfcv 2220 . 2 𝑥𝐵
3 nfv 1462 . 2 𝑦𝜓
4 nfv 1462 . 2 𝑥𝜒
5 cbvralv2.2 . 2 (𝑥 = 𝑦𝐴 = 𝐵)
6 cbvralv2.1 . 2 (𝑥 = 𝑦 → (𝜓𝜒))
71, 2, 3, 4, 5, 6cbvrexcsf 2966 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-sbc 2817  df-csb 2910
This theorem is referenced by: (None)
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