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Theorem rexeqf 3353
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 3352 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥𝐵 ¬ 𝜑))
4 ralnex 3071 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
5 ralnex 3071 . . 3 (∀𝑥𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥𝐵 𝜑)
63, 4, 53bitr3g 313 . 2 (𝐴 = 𝐵 → (¬ ∃𝑥𝐴 𝜑 ↔ ¬ ∃𝑥𝐵 𝜑))
76con4bid 317 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1539  wnfc 2889  wral 3060  wrex 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070
This theorem is referenced by:  rexeqbid  3356  reueq1f  3424  zfrep6  7980  iuneq12daf  32570  indexa  37741  rexeqif  45176
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