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Theorem rexprg 4597
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 2145, ax-12 2179. (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 321 . . 3 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
3 ralprg.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
43notbid 321 . . 3 (𝑥 = 𝐵 → (¬ 𝜑 ↔ ¬ 𝜒))
52, 4ralprg 4595 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒)))
6 ralnex 3150 . . . 4 (∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑)
7 pm4.56 988 . . . 4 ((¬ 𝜓 ∧ ¬ 𝜒) ↔ ¬ (𝜓𝜒))
86, 7bibi12i 343 . . 3 ((∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒)) ↔ (¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ¬ (𝜓𝜒)))
9 notbi 322 . . 3 ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)) ↔ (¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ¬ (𝜓𝜒)))
108, 9sylbb2 241 . 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 209  wa 399  wo 846   = wceq 1542  wcel 2114  wral 3054  wrex 3055  {cpr 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3402  df-un 3858  df-sn 4527  df-pr 4529
This theorem is referenced by:  rextpg  4600  rexpr  4602  reurexprg  4605  fr2nr  5513  sgrp2nmndlem5  18222  nb3grprlem2  27335  nfrgr2v  28221  3vfriswmgrlem  28226  brfvrcld  40885  rnmptpr  42291  ldepspr  45395  zlmodzxzldeplem4  45425
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