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Theorem rextp 4634
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 4627 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
81, 2, 3, 7mp3an 1452 1 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3o 1078   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  {ctp 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-sbc 3770  df-un 3938  df-sn 4558  df-pr 4560  df-tp 4562
This theorem is referenced by:  1cubr  25347
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