ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rextpg GIF version

Theorem rextpg 3524
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 3522 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
43orbi1d 746 . . . 4 ((𝐴𝑉𝐵𝑊) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑)))
5 raltpg.3 . . . . . 6 (𝑥 = 𝐶 → (𝜑𝜃))
65rexsng 3512 . . . . 5 (𝐶𝑋 → (∃𝑥 ∈ {𝐶}𝜑𝜃))
76orbi2d 745 . . . 4 (𝐶𝑋 → (((𝜓𝜒) ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
84, 7sylan9bb 453 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
983impa 1144 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∨ 𝜃)))
10 df-tp 3482 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
1110rexeqi 2589 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
12 rexun 3203 . . 3 (∃𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
1311, 12bitri 183 . 2 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ∨ ∃𝑥 ∈ {𝐶}𝜑))
14 df-3or 931 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∨ 𝜃))
159, 13, 143bitr4g 222 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 670  w3o 929  w3a 930   = wceq 1299  wcel 1448  wrex 2376  cun 3019  {csn 3474  {cpr 3475  {ctp 3476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-sn 3480  df-pr 3481  df-tp 3482
This theorem is referenced by:  rextp  3528
  Copyright terms: Public domain W3C validator