Theorem ifnotdc 3563

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 831	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		notnot 625	. . . . 5 |- ( ph -> -. -. ph )
3	2	iffalsed 3537	. . . 4 |- ( ph -> if ( -. ph , A , B ) = B )
4		iftrue 3532	. . . 4 |- ( ph -> if ( ph , B , A ) = B )
5	3, 4	eqtr4d 2207	. . 3 |- ( ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
6		iftrue 3532	. . . 4 |- ( -. ph -> if ( -. ph , A , B ) = A )
7		iffalse 3535	. . . 4 |- ( -. ph -> if ( ph , B , A ) = A )
8	6, 7	eqtr4d 2207	. . 3 |- ( -. ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
9	5, 8	jaoi 712	. 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 704  DECID wdc 830   = wceq 1349   ifcif 3527

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-11 1500  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-dc 831  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-if 3528

This theorem is referenced by:  lgsneg  13804  lgsdilem  13807