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Theorem rextpg 4615
Description: Convert a restricted existential 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 4612 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
43orbi1d 917 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑)))
5 raltpg.3 . . . . . 6 (𝑥 = 𝐶 → (𝜑𝜃))
65rexsng 4590 . . . . 5 (𝐶𝑋 → (∃𝑥 ∈ {𝐶}𝜑𝜃))
76orbi2d 916 . . . 4 (𝐶𝑋 → (((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
84, 7sylan9bb 513 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
983impa 1112 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
10 df-tp 4546 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110rexeqi 3324 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
12 rexun 4104 . . 3 (∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
1311, 12bitri 278 . 2 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
14 df-3or 1090 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∨ 𝜃))
159, 13, 143bitr4g 317 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847  w3o 1088  w3a 1089   = wceq 1543  wcel 2110  wrex 3062  cun 3864  {csn 4541  {cpr 4543  {ctp 4545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-v 3410  df-un 3871  df-sn 4542  df-pr 4544  df-tp 4546
This theorem is referenced by:  rextp  4622  fr3nr  7557  nb3grprlem2  27469  frgr3vlem2  28357  3vfriswmgr  28361
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