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Theorem ifpr 4688
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 3485 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3485 . 2 (𝐵𝐷𝐵 ∈ V)
3 ifcl 4566 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V)
4 ifeqor 4572 . . . 4 (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
5 elprg 4642 . . . 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 595 1 ((𝐴𝐶𝐵𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 844   = wceq 1533  wcel 2098  Vcvv 3466  ifcif 4521  {cpr 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-if 4522  df-sn 4622  df-pr 4624
This theorem is referenced by:  suppr  9463  infpr  9495  uvcvvcl  21652  indf  33505
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