Theorem ifeq2dadc 3637

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

Hypotheses

Ref	Expression
ifeq2da.1	|- ( ( ph /\ -. ps ) -> A = B )
ifeq2dadc.dc	|- ( ph -> DECID ps )

Assertion

Ref	Expression
ifeq2dadc	|- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )

Proof of Theorem ifeq2dadc

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ph /\ ps ) -> ps )
2	1	iftrued 3612	. . 3 |- ( ( ph /\ ps ) -> if ( ps , C , A ) = C )
3	1	iftrued 3612	. . 3 |- ( ( ph /\ ps ) -> if ( ps , C , B ) = C )
4	2, 3	eqtr4d 2267	. 2 |- ( ( ph /\ ps ) -> if ( ps , C , A ) = if ( ps , C , B ) )
5		ifeq2da.1	. . 3 |- ( ( ph /\ -. ps ) -> A = B )
6	5	ifeq2d 3624	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , C , A ) = if ( ps , C , B ) )
7		ifeq2dadc.dc	. . 3 |- ( ph -> DECID ps )
8		exmiddc 843	. . 3 |- (DECID ps -> ( ps \/ -. ps ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( ps \/ -. ps ) )
10	4, 6, 9	mpjaodan 805	1 |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   = wceq 1397   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-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  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-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-if 3606

This theorem is referenced by:  subgmulg  13774