Theorem iffalse 3610

Description: Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)

Assertion

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

Proof of Theorem iffalse

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-if 3603	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		dedlemb 976	. . 3 |- ( -. ph -> ( x e. B <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) ) )
3	2	abbi2dv 2348	. 2 |- ( -. ph -> B = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } )
4	1, 3	eqtr4id 2281	1 |- ( -. ph -> if ( ph , A , B ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {cab 2215   ifcif 3602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603

This theorem is referenced by:  iffalsei  3611  iffalsed  3612  ifnefalse  3613  ifsbdc  3615  ifcldadc  3632  ifeq1dadc  3633  ifeqdadc  3635  ifbothdadc  3636  ifbothdc  3637  ifiddc  3638  ifcldcd  3640  ifnotdc  3641  2if2dc  3642  ifandc  3643  ifordc  3644  ifnetruedc  3646  pw2f1odclem  7003  fidifsnen  7040  nnnninf  7304  uzin  9767  modifeq2int  10620  seqf1oglem1  10753  seqf1oglem2  10754  bcval  10983  bcval3  10985  swrdccat  11282  pfxccat3a  11285  swrdccat3b  11287  sumrbdclem  11903  fsum3cvg  11904  summodclem2a  11907  sumsplitdc  11958  prodrbdclem  12097  fproddccvg  12098  prodssdc  12115  flodddiv4  12462  gcdn0val  12497  dfgcd2  12550  lcmn0val  12603  pcgcd  12867  pcmptcl  12880  pcmpt  12881  pcmpt2  12882  pcprod  12884  fldivp1  12886  unct  13028  lgsneg  15718  lgsdilem  15721  lgsdir2  15727  lgsdir  15729  lgsdi  15731  lgsne0  15732  gausslemma2dlem1a  15752  2lgslem1c  15784  2lgs  15798