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Theorem ifexg 4578
Description: Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)
Assertion
Ref Expression
ifexg ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Proof of Theorem ifexg
StepHypRef Expression
1 simpl 482 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpr 484 . 2 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
31, 2ifexd 4577 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  Vcvv 3471  ifcif 4529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-if 4530
This theorem is referenced by:  fsuppmptif  9422  cantnfp1lem1  9701  cantnfp1lem3  9703  symgextfv  19372  pmtrfv  19406  marrepeval  22464  gsummatr01lem3  22558  stdbdmetval  24422  stdbdxmet  24423  ellimc2  25805  cdleme31fv  39863
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