Theorem iftrue 3474

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		dedlema 953	. . 3 |- ( ph -> ( x e. A <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) ) )
2	1	abbi2dv 2256	. 2 |- ( ph -> A = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } )
3		df-if 3470	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
4	2, 3	syl6reqr 2189	1 |- ( ph -> if ( ph , A , B ) = A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   {cab 2123   ifcif 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-if 3470

This theorem is referenced by:  iftruei  3475  iftrued  3476  ifsbdc  3481  ifcldadc  3496  ifbothdadc  3498  ifbothdc  3499  ifiddc  3500  ifcldcd  3502  ifandc  3503  fidifsnen  6757  nnnninf  7016  mkvprop  7025  uzin  9351  fzprval  9855  fztpval  9856  modifeq2int  10152  bcval  10488  bcval2  10489  sumrbdclem  11138  fsum3cvg  11139  summodclem2a  11143  isumss2  11155  fsum3ser  11159  fsumsplit  11169  sumsplitdc  11194  prodrbdclem  11333  fproddccvg  11334  flodddiv4  11620  gcd0val  11638  dfgcd2  11691  eucalgf  11725  eucalginv  11726  eucalglt  11727  unct  11943  dvexp2  12834  nnsf  13188  nninfsellemsuc  13197