Theorem iftrue 3668

Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
iftrue	|- (ph -> if(ph, A, B) = A)

Proof of Theorem iftrue

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedlem0a 918	. . 3 |- (ph -> (x e. A <-> ((x e. B -> ph) -> (x e. A /\ ph))))
2	1	abbi2dv 2468	. 2 |- (ph -> A = {x | ((x e. B -> ph) -> (x e. A /\ ph))})
3		dfif2 3664	. 2 |- if(ph, A, B) = {x | ((x e. B -> ph) -> (x e. A /\ ph))}
4	2, 3	syl6reqr 2404	1 |- (ph -> if(ph, A, B) = A)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  ifcif 3662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663

This theorem is referenced by:  ifsb  3671  ifbi  3679  ifeq2da  3688  ifclda  3689  elimif  3691  ifbothda  3692  ifid  3694  ifeqor  3699  ifnot  3700  ifan  3701  ifor  3702  dedth  3703  elimhyp  3710  elimhyp2v  3711  elimhyp3v  3712  elimhyp4v  3713  elimdhyp  3715  keephyp2v  3717  keephyp3v  3718  setswith  4321  dfiota3  4370  eqtfinrelk  4486  tfinnul  4491  dfphi2  4569  phi11lem1  4595  0cnelphi  4597  phialllem1  4616  elimdelov  5573  enprmaplem5  6080