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Theorem rextpg 3424
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 3422 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
43orbi1d 705 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑)))
5 raltpg.3 . . . . . 6 (𝑥 = 𝐶 → (𝜑𝜃))
65rexsng 3412 . . . . 5 (𝐶𝑋 → (∃𝑥 ∈ {𝐶}𝜑𝜃))
76orbi2d 704 . . . 4 (𝐶𝑋 → (((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
84, 7sylan9bb 435 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
983impa 1099 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
10 df-tp 3383 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110rexeqi 2510 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
12 rexun 3123 . . 3 (∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
1311, 12bitri 173 . 2 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
14 df-3or 886 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∨ 𝜃))
159, 13, 143bitr4g 212 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wo 629  w3o 884  w3a 885   = wceq 1243  wcel 1393  wrex 2307  cun 2915  {csn 3375  {cpr 3376  {ctp 3377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-tp 3383
This theorem is referenced by:  rextp  3428
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