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Theorem rexpr 4230
Description: Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralpr.1 𝐴 ∈ V
ralpr.2 𝐵 ∈ V
ralpr.3 (𝑥 = 𝐴 → (𝜑𝜓))
ralpr.4 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
rexpr (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexpr
StepHypRef Expression
1 ralpr.1 . 2 𝐴 ∈ V
2 ralpr.2 . 2 𝐵 ∈ V
3 ralpr.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 ralpr.4 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
53, 4rexprg 4226 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
61, 2, 5mp2an 707 1 (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383   = wceq 1481  wcel 1988  wrex 2910  Vcvv 3195  {cpr 4170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-v 3197  df-sbc 3430  df-un 3572  df-sn 4169  df-pr 4171
This theorem is referenced by:  xpsdsval  22167  poimir  33413
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