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  6991  fidifsnen  7028  nnnninf  7289  uzin  9751  modifeq2int  10603  seqf1oglem1  10736  seqf1oglem2  10737  bcval  10966  bcval3  10968  swrdccat  11262  pfxccat3a  11265  swrdccat3b  11267  sumrbdclem  11883  fsum3cvg  11884  summodclem2a  11887  sumsplitdc  11938  prodrbdclem  12077  fproddccvg  12078  prodssdc  12095  flodddiv4  12442  gcdn0val  12477  dfgcd2  12530  lcmn0val  12583  pcgcd  12847  pcmptcl  12860  pcmpt  12861  pcmpt2  12862  pcprod  12864  fldivp1  12866  unct  13008  lgsneg  15697  lgsdilem  15700  lgsdir2  15706  lgsdir  15708  lgsdi  15710  lgsne0  15711  gausslemma2dlem1a  15731  2lgslem1c  15763  2lgs  15777