Theorem ifandc 3643

Description: Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)

Assertion

Ref	Expression
ifandc	|- (DECID ph -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )

Proof of Theorem ifandc

Step	Hyp	Ref	Expression
1		df-dc 840	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		iftrue 3607	. . . 4 |- ( ph -> if ( ph , if ( ps , A , B ) , B ) = if ( ps , A , B ) )
3		ibar 301	. . . . 5 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
4	3	ifbid 3624	. . . 4 |- ( ph -> if ( ps , A , B ) = if ( ( ph /\ ps ) , A , B ) )
5	2, 4	eqtr2d 2263	. . 3 |- ( ph -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )
6		simpl 109	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
7	6	con3i 635	. . . . 5 |- ( -. ph -> -. ( ph /\ ps ) )
8	7	iffalsed 3612	. . . 4 |- ( -. ph -> if ( ( ph /\ ps ) , A , B ) = B )
9		iffalse 3610	. . . 4 |- ( -. ph -> if ( ph , if ( ps , A , B ) , B ) = B )
10	8, 9	eqtr4d 2265	. . 3 |- ( -. ph -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )
11	5, 10	jaoi 721	. 2 |- ( ( ph \/ -. ph ) -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )
12	1, 11	sylbi 121	1 |- (DECID ph -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   ifcif 3602

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603

This theorem is referenced by:  isumss  11902