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Theorem rexeqf 3358
Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1 𝑥𝐴
raleq1f.2 𝑥𝐵
Assertion
Ref Expression
rexeqf (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Proof of Theorem rexeqf
StepHypRef Expression
1 raleq1f.1 . . . 4 𝑥𝐴
2 raleq1f.2 . . . 4 𝑥𝐵
31, 2nfeq 2960 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2871 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 629 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5exbid 2190 . 2 (𝐴 = 𝐵 → (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝑥𝐵𝜑)))
7 df-rex 3111 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 3111 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 315 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081  wnfc 2933  wrex 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rex 3111
This theorem is referenced by:  rexeqOLD  3370  rexeqbid  3382  zfrep6  7512  iuneq12daf  29998  indexa  34559
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