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Theorem rexprg 4182
 Description: Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
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 df-pr 4128 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
21rexeqi 3120 . . 3 (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3 rexun 3755 . . 3 (∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑))
42, 3bitri 263 . 2 (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑))
5 ralprg.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65rexsng 4166 . . . 4 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
76orbi1d 735 . . 3 (𝐴𝑉 → ((∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ ∃𝑥 ∈ {𝐵}𝜑)))
8 ralprg.2 . . . . 5 (𝑥 = 𝐵 → (𝜑𝜒))
98rexsng 4166 . . . 4 (𝐵𝑊 → (∃𝑥 ∈ {𝐵}𝜑𝜒))
109orbi2d 734 . . 3 (𝐵𝑊 → ((𝜓 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓𝜒)))
117, 10sylan9bb 732 . 2 ((𝐴𝑉𝐵𝑊) → ((∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓𝜒)))
124, 11syl5bb 271 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∪ cun 3538  {csn 4125  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  rextpg  4184  rexpr  4186  fr2nr  5016  sgrp2nmndlem5  17239  nb3graprlem2  25981  frgra2v  26526  3vfriswmgralem  26531  brfvrcld  37002  rnmptpr  38353  nb3grprlem2  40609  nfrgr2v  41442  3vfriswmgrlem  41447  ldepspr  42056  zlmodzxzldeplem4  42086
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