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Theorem rexeqi 2563
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 2559 . 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 1287  wrex 2356
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361
This theorem is referenced by:  rexrab2  2773  rexprg  3479  rextpg  3481  rexxp  4550  rexrnmpt2  5719  arch  8606  infssuzex  10851  gcdsupex  10855  gcdsupcl  10856
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