Theorem iffalse 3482

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		dedlemb 954	. . 3 |- ( -. ph -> ( x e. B <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) ) )
2	1	abbi2dv 2258	. 2 |- ( -. ph -> B = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } )
3		df-if 3475	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
4	2, 3	syl6reqr 2191	1 |- ( -. ph -> if ( ph , A , B ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   {cab 2125   ifcif 3474

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 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475

This theorem is referenced by:  iffalsei  3483  iffalsed  3484  ifnefalse  3485  ifsbdc  3486  ifcldadc  3501  ifeq1dadc  3502  ifbothdadc  3503  ifbothdc  3504  ifiddc  3505  ifcldcd  3507  ifandc  3508  fidifsnen  6764  nnnninf  7023  uzin  9358  modifeq2int  10159  bcval  10495  bcval3  10497  sumrbdclem  11146  fsum3cvg  11147  summodclem2a  11150  sumsplitdc  11201  prodrbdclem  11340  fproddccvg  11341  flodddiv4  11631  gcdn0val  11650  dfgcd2  11702  lcmn0val  11747  unct  11954