Theorem iffalse 3613

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 3606	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		dedlemb 978	. . 3 |- ( -. ph -> ( x e. B <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) ) )
3	2	abbi2dv 2350	. 2 |- ( -. ph -> B = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } )
4	1, 3	eqtr4id 2283	1 |- ( -. ph -> if ( ph , A , B ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   {cab 2217   ifcif 3605

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 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606

This theorem is referenced by:  iffalsei  3614  iffalsed  3615  ifnefalse  3616  ifsbdc  3618  ifcldadc  3635  ifeq1dadc  3636  ifeqdadc  3638  ifbothdadc  3639  ifbothdc  3640  ifiddc  3641  ifcldcd  3643  ifnotdc  3644  2if2dc  3645  ifandc  3646  ifordc  3647  ifnetruedc  3649  pw2f1odclem  7020  fidifsnen  7057  nnnninf  7325  uzin  9789  modifeq2int  10648  seqf1oglem1  10781  seqf1oglem2  10782  bcval  11011  bcval3  11013  swrdccat  11316  pfxccat3a  11319  swrdccat3b  11321  sumrbdclem  11939  fsum3cvg  11940  summodclem2a  11943  sumsplitdc  11994  prodrbdclem  12133  fproddccvg  12134  prodssdc  12151  flodddiv4  12498  gcdn0val  12533  dfgcd2  12586  lcmn0val  12639  pcgcd  12903  pcmptcl  12916  pcmpt  12917  pcmpt2  12918  pcprod  12920  fldivp1  12922  unct  13064  lgsneg  15755  lgsdilem  15758  lgsdir2  15764  lgsdir  15766  lgsdi  15768  lgsne0  15769  gausslemma2dlem1a  15789  2lgslem1c  15821  2lgs  15835