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  7019  fidifsnen  7056  nnnninf  7324  uzin  9788  modifeq2int  10647  seqf1oglem1  10780  seqf1oglem2  10781  bcval  11010  bcval3  11012  swrdccat  11315  pfxccat3a  11318  swrdccat3b  11320  sumrbdclem  11937  fsum3cvg  11938  summodclem2a  11941  sumsplitdc  11992  prodrbdclem  12131  fproddccvg  12132  prodssdc  12149  flodddiv4  12496  gcdn0val  12531  dfgcd2  12584  lcmn0val  12637  pcgcd  12901  pcmptcl  12914  pcmpt  12915  pcmpt2  12916  pcprod  12918  fldivp1  12920  unct  13062  lgsneg  15752  lgsdilem  15755  lgsdir2  15761  lgsdir  15763  lgsdi  15765  lgsne0  15766  gausslemma2dlem1a  15786  2lgslem1c  15818  2lgs  15832