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Theorem rexeqdv 2693
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
Hypothesis
Ref Expression
raleq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
rexeqdv (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rexeqdv
StepHypRef Expression
1 raleq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 rexeq 2687 . 2 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
31, 2syl 14 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wrex 2469
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474
This theorem is referenced by:  rexeqbidv  2699  rexeqbidva  2701  fnunirn  5784  cbvexfo  5803  fival  6987  nninfwlpoimlemg  7191  nninfwlpoimlemginf  7192  nninfwlpoim  7194  genipv  7526  exfzdc  10258  zproddc  11605  infssuzex  11968  nninfdcex  11972  ennnfonelemrnh  12435  grppropd  12928  dvdsrpropdg  13458  cnpfval  14092
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