Theorem ifnotdc 3568

Description: Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)

Assertion

Ref	Expression
ifnotdc	|- (DECID ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )

Proof of Theorem ifnotdc

Step	Hyp	Ref	Expression
1		df-dc 835	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		notnot 629	. . . . 5 |- ( ph -> -. -. ph )
3	2	iffalsed 3542	. . . 4 |- ( ph -> if ( -. ph , A , B ) = B )
4		iftrue 3537	. . . 4 |- ( ph -> if ( ph , B , A ) = B )
5	3, 4	eqtr4d 2211	. . 3 |- ( ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
6		iftrue 3537	. . . 4 |- ( -. ph -> if ( -. ph , A , B ) = A )
7		iffalse 3540	. . . 4 |- ( -. ph -> if ( ph , B , A ) = A )
8	6, 7	eqtr4d 2211	. . 3 |- ( -. ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
9	5, 8	jaoi 716	. 2 |- ( ( ph \/ -. ph ) -> if ( -. ph , A , B ) = if ( ph , B , A ) )
10	1, 9	sylbi 121	1 |- (DECID ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708  DECID wdc 834   = wceq 1353   ifcif 3532

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-dc 835  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-if 3533

This theorem is referenced by:  lgsneg  13996  lgsdilem  13999