Theorem ifnotdc 3618

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 837	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		notnot 630	. . . . 5 |- ( ph -> -. -. ph )
3	2	iffalsed 3589	. . . 4 |- ( ph -> if ( -. ph , A , B ) = B )
4		iftrue 3584	. . . 4 |- ( ph -> if ( ph , B , A ) = B )
5	3, 4	eqtr4d 2243	. . 3 |- ( ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
6		iftrue 3584	. . . 4 |- ( -. ph -> if ( -. ph , A , B ) = A )
7		iffalse 3587	. . . 4 |- ( -. ph -> if ( ph , B , A ) = A )
8	6, 7	eqtr4d 2243	. . 3 |- ( -. ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
9	5, 8	jaoi 718	. 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 710  DECID wdc 836   = wceq 1373   ifcif 3579

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-dc 837  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-if 3580

This theorem is referenced by:  lgsneg  15616  lgsdilem  15619