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Theorem rextpg 4184
 Description: Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1 (𝑥 = 𝐴 → (𝜑𝜓))
ralprg.2 (𝑥 = 𝐵 → (𝜑𝜒))
raltpg.3 (𝑥 = 𝐶 → (𝜑𝜃))
Assertion
Ref Expression
rextpg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem rextpg
StepHypRef Expression
1 ralprg.1 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
2 ralprg.2 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜒))
31, 2rexprg 4182 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
43orbi1d 735 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑)))
5 raltpg.3 . . . . . 6 (𝑥 = 𝐶 → (𝜑𝜃))
65rexsng 4166 . . . . 5 (𝐶𝑋 → (∃𝑥 ∈ {𝐶}𝜑𝜃))
76orbi2d 734 . . . 4 (𝐶𝑋 → (((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
84, 7sylan9bb 732 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
983impa 1251 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
10 df-tp 4130 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110rexeqi 3120 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
12 rexun 3755 . . 3 (∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
1311, 12bitri 263 . 2 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
14 df-3or 1032 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∨ 𝜃))
159, 13, 143bitr4g 302 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∪ cun 3538  {csn 4125  {cpr 4127  {ctp 4129 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-3or 1032  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  df-tp 4130 This theorem is referenced by:  rextp  4188  fr3nr  6871  nb3graprlem2  25981  frgra3vlem2  26528  3vfriswmgra  26532  nb3grprlem2  40609  frgr3vlem2  41444  3vfriswmgr  41448
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