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Theorem ifexg 4570
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 4569 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  Vcvv 3466  ifcif 4521
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-if 4522
This theorem is referenced by:  fsuppmptif  9391  cantnfp1lem1  9670  cantnfp1lem3  9672  symgextfv  19334  pmtrfv  19368  marrepeval  22409  gsummatr01lem3  22503  stdbdmetval  24367  stdbdxmet  24368  ellimc2  25750  cdleme31fv  39764
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