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Theorem cbvrexv2 3136
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 2329 . 2 𝑦𝐴
2 nfcv 2329 . 2 𝑥𝐵
3 nfv 1538 . 2 𝑦𝜓
4 nfv 1538 . 2 𝑥𝜒
5 cbvralv2.2 . 2 (𝑥 = 𝑦𝐴 = 𝐵)
6 cbvralv2.1 . 2 (𝑥 = 𝑦 → (𝜓𝜒))
71, 2, 3, 4, 5, 6cbvrexcsf 3132 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1363  wrex 2466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-sbc 2975  df-csb 3070
This theorem is referenced by: (None)
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