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Theorem ifpr 4657
Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
Assertion
Ref Expression
ifpr ((𝐴𝐶𝐵𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})

Proof of Theorem ifpr
StepHypRef Expression
1 elex 3466 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3466 . 2 (𝐵𝐷𝐵 ∈ V)
3 ifcl 4536 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V)
4 ifeqor 4542 . . . 4 (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
5 elprg 4612 . . . 4 (if(𝜑, 𝐴, 𝐵) ∈ V → (if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵} ↔ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)))
64, 5mpbiri 258 . . 3 (if(𝜑, 𝐴, 𝐵) ∈ V → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
73, 6syl 17 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
81, 2, 7syl2an 597 1 ((𝐴𝐶𝐵𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  Vcvv 3448  ifcif 4491  {cpr 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-un 3920  df-if 4492  df-sn 4592  df-pr 4594
This theorem is referenced by:  suppr  9414  infpr  9446  uvcvvcl  21209  indf  32654
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