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Theorem rexprg 4631
 Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 8-Apr-2023.)
Hypotheses
Ref Expression
ralprg.1 (𝑥 = 𝐴 → (𝜑𝜓))
ralprg.2 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
rexprg ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem rexprg
StepHypRef Expression
1 nfv 1908 . 2 𝑥𝜓
2 nfv 1908 . 2 𝑥𝜒
3 ralprg.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 ralprg.2 . 2 (𝑥 = 𝐵 → (𝜑𝜒))
51, 2, 3, 4rexprgf 4629 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 843   = wceq 1530   ∈ wcel 2106  ∃wrex 3143  {cpr 4565 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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-rex 3148  df-v 3501  df-sbc 3776  df-un 3944  df-sn 4564  df-pr 4566 This theorem is referenced by:  rextpg  4633  rexpr  4635  reurexprg  4638  fr2nr  5531  sgrp2nmndlem5  18029  nb3grprlem2  27078  nfrgr2v  27966  3vfriswmgrlem  27971  brfvrcld  39898  rnmptpr  41295  ldepspr  44357  zlmodzxzldeplem4  44387
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