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Theorem cbvrex 2693
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvral.1 𝑦𝜑
cbvral.2 𝑥𝜓
cbvral.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrex (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvrex
StepHypRef Expression
1 nfcv 2312 . 2 𝑥𝐴
2 nfcv 2312 . 2 𝑦𝐴
3 cbvral.1 . 2 𝑦𝜑
4 cbvral.2 . 2 𝑥𝜓
5 cbvral.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvrexf 2690 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1453  wrex 2449
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454
This theorem is referenced by:  cbvrmo  2695  cbvrexv  2697  cbvrexsv  2713  cbviun  3908  disjiun  3982  rexxpf  4756  isarep1  5282  rexrnmpt  5636  elabrex  5734
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