Theorem ifeq1dadc 3671

Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
ifeq1dadc.1	|- ( ( ph /\ ps ) -> A = B )
ifeq1dadc.dc	|- ( ph -> DECID ps )

Assertion

Ref	Expression
ifeq1dadc	|- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

Proof of Theorem ifeq1dadc

Step	Hyp	Ref	Expression
1		ifeq1dadc.1	. . 3 |- ( ( ph /\ ps ) -> A = B )
2	1	ifeq1d 3658	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
3		iffalse 3648	. . . 4 |- ( -. ps -> if ( ps , A , C ) = C )
4		iffalse 3648	. . . 4 |- ( -. ps -> if ( ps , B , C ) = C )
5	3, 4	eqtr4d 2274	. . 3 |- ( -. ps -> if ( ps , A , C ) = if ( ps , B , C ) )
6	5	adantl 277	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , C ) = if ( ps , B , C ) )
7		ifeq1dadc.dc	. . 3 |- ( ph -> DECID ps )
8		exmiddc 848	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
10	2, 6, 9	mpjaodan 810	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   ifcif 3638

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3639

This theorem is used by:  sumeq2  12125  isumss  12158  prodeq2  12324  lgsval2lem  16129  lgsval4lem  16130  lgsneg  16143  lgsmod  16145  lgsdilem2  16155