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Theorem rexeqf 3350
Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3321 for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 9-Mar-2025.)
Hypotheses
Ref Expression
raleqf.1 𝑥𝐴
raleqf.2 𝑥𝐵
Assertion
Ref Expression
rexeqf (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Proof of Theorem rexeqf
StepHypRef Expression
1 raleqf.1 . . . 4 𝑥𝐴
2 raleqf.2 . . . 4 𝑥𝐵
31, 2raleqf 3349 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥𝐵 ¬ 𝜑))
4 ralnex 3072 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
5 ralnex 3072 . . 3 (∀𝑥𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐵 𝜑)
63, 4, 53bitr3g 312 . 2 (𝐴 = 𝐵 → (¬ ∃𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐵 𝜑))
76con4bid 316 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wnfc 2883  wral 3061  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071
This theorem is referenced by:  rexeqbid  3353  reueq1f  3421  zfrep6  7937  iuneq12daf  31775  indexa  36589  rexeqif  43846
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