Theorem iftrue 1780

Description: Value of the conditional operator when its first argument is true.

Assertion

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

Proof of Theorem iftrue

Step	Hyp	Ref	Expression
1		dedlema 569	. . 3 |- (ph -> (x e. A <-> ((x e. A /\ ph) \/ (x e. B /\ -. ph))))
2	1	biabrdv 1184	. 2 |- (ph -> A = {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))})
3		df-if 1777	. 2 |- if(ph, A, B) = {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))}
4	2, 3	syl6reqr 1143	1 |- (ph -> if(ph, A, B) = A)

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  ifcif 1776

This theorem is referenced by:  ifbi 1783  dedth 1784  dedth2v 1785  dedth3v 1786  elimhyp 1790  elimhyp2v 1791  elimhyp3v 1792  keephyp 1794  keephyp3v 1795  oe0m 3127  unxpdomlem 3649  ruclem13 4897  ruclem18 4902  ruclem19 4903  znnen 4930

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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