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Theorem rextpg 3661
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 3659 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
43orbi1d 792 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑)))
5 raltpg.3 . . . . . 6 (𝑥 = 𝐶 → (𝜑𝜃))
65rexsng 3648 . . . . 5 (𝐶𝑋 → (∃𝑥 ∈ {𝐶}𝜑𝜃))
76orbi2d 791 . . . 4 (𝐶𝑋 → (((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
84, 7sylan9bb 462 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
983impa 1196 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
10 df-tp 3615 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110rexeqi 2691 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
12 rexun 3330 . . 3 (∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
1311, 12bitri 184 . 2 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
14 df-3or 981 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∨ 𝜃))
159, 13, 143bitr4g 223 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709  w3o 979  w3a 980   = wceq 1364  wcel 2160  wrex 2469  cun 3142  {csn 3607  {cpr 3608  {ctp 3609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-sn 3613  df-pr 3614  df-tp 3615
This theorem is referenced by:  rextp  3665
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