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Theorem rextp 3458
 Description: Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
raltp.1 𝐴 ∈ V
raltp.2 𝐵 ∈ V
raltp.3 𝐶 ∈ V
raltp.4 (𝑥 = 𝐴 → (𝜑𝜓))
raltp.5 (𝑥 = 𝐵 → (𝜑𝜒))
raltp.6 (𝑥 = 𝐶 → (𝜑𝜃))
Assertion
Ref Expression
rextp (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rextp
StepHypRef Expression
1 raltp.1 . 2 𝐴 ∈ V
2 raltp.2 . 2 𝐵 ∈ V
3 raltp.3 . 2 𝐶 ∈ V
4 raltp.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
5 raltp.5 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
6 raltp.6 . . 3 (𝑥 = 𝐶 → (𝜑𝜃))
74, 5, 6rextpg 3454 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
81, 2, 3, 7mp3an 1269 1 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   ∨ w3o 919   = wceq 1285   ∈ wcel 1434  ∃wrex 2350  Vcvv 2602  {ctp 3408 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3412  df-pr 3413  df-tp 3414 This theorem is referenced by: (None)
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