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Theorem ifexg 4129
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 4062 . . 3 (𝑥 = 𝐴 → if(𝜑, 𝑥, 𝑦) = if(𝜑, 𝐴, 𝑦))
21eleq1d 2683 . 2 (𝑥 = 𝐴 → (if(𝜑, 𝑥, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝑦) ∈ V))
3 ifeq2 4063 . . 3 (𝑦 = 𝐵 → if(𝜑, 𝐴, 𝑦) = if(𝜑, 𝐴, 𝐵))
43eleq1d 2683 . 2 (𝑦 = 𝐵 → (if(𝜑, 𝐴, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝐵) ∈ V))
5 vex 3189 . . 3 𝑥 ∈ V
6 vex 3189 . . 3 𝑦 ∈ V
75, 6ifex 4128 . 2 if(𝜑, 𝑥, 𝑦) ∈ V
82, 4, 7vtocl2g 3256 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  ifcif 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-un 3560  df-if 4059
This theorem is referenced by:  fsuppmptif  8249  cantnfp1lem1  8519  cantnfp1lem3  8521  symgextfv  17759  pmtrfv  17793  evlslem3  19433  marrepeval  20288  gsummatr01lem3  20382  stdbdmetval  22229  stdbdxmet  22230  ellimc2  23547  psgnfzto1stlem  29632  cdleme31fv  35155  sge0val  39887  hsphoival  40097  hspmbllem2  40145
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