MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifexg Structured version   Visualization version   GIF version

Theorem ifexg 4536
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 484 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpr 486 . 2 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
31, 2ifexd 4535 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3444  ifcif 4487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-if 4488
This theorem is referenced by:  fsuppmptif  9340  cantnfp1lem1  9619  cantnfp1lem3  9621  symgextfv  19205  pmtrfv  19239  marrepeval  21928  gsummatr01lem3  22022  stdbdmetval  23886  stdbdxmet  23887  ellimc2  25257  cdleme31fv  38899
  Copyright terms: Public domain W3C validator