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Theorem rexeqfOLD 3345
Description: Obsolete version of rexeqf 3344 as of 9-Mar-2025. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
raleqf.1 𝑥𝐴
raleqf.2 𝑥𝐵
Assertion
Ref Expression
rexeqfOLD (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Proof of Theorem rexeqfOLD
StepHypRef Expression
1 raleqf.1 . . . 4 𝑥𝐴
2 raleqf.2 . . . 4 𝑥𝐵
31, 2nfeq 2910 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2816 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 629 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5exbid 2208 . 2 (𝐴 = 𝐵 → (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝑥𝐵𝜑)))
7 df-rex 3065 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 3065 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 314 1 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wnfc 2877  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rex 3065
This theorem is referenced by: (None)
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