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Theorem ifexg 3721
 Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
Assertion
Ref Expression
ifexg ((A V B W) → 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 3666 . . 3 (x = A → if(φ, x, y) = if(φ, A, y))
21eleq1d 2419 . 2 (x = A → ( if(φ, x, y) V ↔ if(φ, A, y) V))
3 ifeq2 3667 . . 3 (y = B → if(φ, A, y) = if(φ, A, B))
43eleq1d 2419 . 2 (y = B → ( if(φ, A, y) V ↔ if(φ, A, B) V))
5 vex 2862 . . 3 x V
6 vex 2862 . . 3 y V
75, 6ifex 3720 . 2 if(φ, x, y) V
82, 4, 7vtocl2g 2918 1 ((A V B W) → if(φ, A, B) V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663 This theorem is referenced by: (None)
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