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Theorem rexeqi 2555
Description: Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
raleq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
rexeqi (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexeqi
StepHypRef Expression
1 raleq1i.1 . 2 𝐴 = 𝐵
2 rexeq 2551 . 2 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
31, 2ax-mp 7 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)
Colors of variables: wff set class
Syntax hints:  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-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355
This theorem is referenced by:  rexrab2  2760  rexprg  3446  rextpg  3448  rexxp  4502  rexrnmpt2  5641  arch  8341  infssuzex  10478  gcdsupex  10482  gcdsupcl  10483
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