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Theorem rexprg 4632
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.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 30-Sep-2024.)
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 ralprg.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
21notbid 318 . . 3 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
3 ralprg.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
43notbid 318 . . 3 (𝑥 = 𝐵 → (¬ 𝜑 ↔ ¬ 𝜒))
52, 4ralprg 4630 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒)))
6 ralnex 3167 . . . 4 (∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑)
7 pm4.56 986 . . . 4 ((¬ 𝜓 ∧ ¬ 𝜒) ↔ ¬ (𝜓𝜒))
86, 7bibi12i 340 . . 3 ((∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒)) ↔ (¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ¬ (𝜓𝜒)))
9 notbi 319 . . 3 ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)) ↔ (¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ¬ (𝜓𝜒)))
108, 9sylbb2 237 . 2 ((∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒)) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
115, 10syl 17 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  wral 3064  wrex 3065  {cpr 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564
This theorem is referenced by:  rextpg  4635  rexpr  4637  reurexprg  4640  fr2nr  5567  sgrp2nmndlem5  18568  nb3grprlem2  27748  nfrgr2v  28636  3vfriswmgrlem  28641  brfvrcld  41299  rnmptpr  42713  ldepspr  45814  zlmodzxzldeplem4  45844
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