Theorem ifnotdc 3555

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 825	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		notnot 619	. . . . 5 |- ( ph -> -. -. ph )
3	2	iffalsed 3529	. . . 4 |- ( ph -> if ( -. ph , A , B ) = B )
4		iftrue 3524	. . . 4 |- ( ph -> if ( ph , B , A ) = B )
5	3, 4	eqtr4d 2201	. . 3 |- ( ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
6		iftrue 3524	. . . 4 |- ( -. ph -> if ( -. ph , A , B ) = A )
7		iffalse 3527	. . . 4 |- ( -. ph -> if ( ph , B , A ) = A )
8	6, 7	eqtr4d 2201	. . 3 |- ( -. ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
9	5, 8	jaoi 706	. 2 |- ( ( ph \/ -. ph ) -> if ( -. ph , A , B ) = if ( ph , B , A ) )
10	1, 9	sylbi 120	1 |- (DECID ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 698  DECID wdc 824   = wceq 1343   ifcif 3519

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-if 3520

This theorem is referenced by:  lgsneg  13525  lgsdilem  13528