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Theorem rexprg 4625
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 1906 . 2 𝑥𝜓
2 nfv 1906 . 2 𝑥𝜒
3 ralprg.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 ralprg.2 . 2 (𝑥 = 𝐵 → (𝜑𝜒))
51, 2, 3, 4rexprgf 4623 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wrex 3136  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-sbc 3770  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  rextpg  4627  rexpr  4629  reurexprg  4632  fr2nr  5526  sgrp2nmndlem5  18032  nb3grprlem2  27090  nfrgr2v  27978  3vfriswmgrlem  27983  brfvrcld  39914  rnmptpr  41309  ldepspr  44456  zlmodzxzldeplem4  44486
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