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Theorem ifexg 4107
 Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
Assertion
Ref Expression
ifexg ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Proof of Theorem ifexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ifeq1 4040 . . 3 (𝑥 = 𝐴 → if(𝜑, 𝑥, 𝑦) = if(𝜑, 𝐴, 𝑦))
21eleq1d 2672 . 2 (𝑥 = 𝐴 → (if(𝜑, 𝑥, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝑦) ∈ V))
3 ifeq2 4041 . . 3 (𝑦 = 𝐵 → if(𝜑, 𝐴, 𝑦) = if(𝜑, 𝐴, 𝐵))
43eleq1d 2672 . 2 (𝑦 = 𝐵 → (if(𝜑, 𝐴, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝐵) ∈ V))
5 vex 3176 . . 3 𝑥 ∈ V
6 vex 3176 . . 3 𝑦 ∈ V
75, 6ifex 4106 . 2 if(𝜑, 𝑥, 𝑦) ∈ V
82, 4, 7vtocl2g 3243 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037 This theorem is referenced by:  fsuppmptif  8188  cantnfp1lem1  8458  cantnfp1lem3  8460  symgextfv  17661  pmtrfv  17695  evlslem3  19335  marrepeval  20188  gsummatr01lem3  20282  stdbdmetval  22129  stdbdxmet  22130  ellimc2  23447  psgnfzto1stlem  29181  cdleme31fv  34696  sge0val  39259  hsphoival  39469  hspmbllem2  39517
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