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Theorem ifexg 4575
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 4574 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  Vcvv 3480  ifcif 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-if 4526
This theorem is referenced by:  fsuppmptif  9439  cantnfp1lem1  9718  cantnfp1lem3  9720  symgextfv  19436  pmtrfv  19470  marrepeval  22569  gsummatr01lem3  22663  stdbdmetval  24527  stdbdxmet  24528  ellimc2  25912  cdleme31fv  40392
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