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Theorem ifexg 4580
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 4579 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  ifcif 4531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-if 4532
This theorem is referenced by:  fsuppmptif  9437  cantnfp1lem1  9716  cantnfp1lem3  9718  symgextfv  19451  pmtrfv  19485  marrepeval  22585  gsummatr01lem3  22679  stdbdmetval  24543  stdbdxmet  24544  ellimc2  25927  cdleme31fv  40373
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