Theorem ifnotdc 3644

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 842	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		notnot 634	. . . . 5 |- ( ph -> -. -. ph )
3	2	iffalsed 3615	. . . 4 |- ( ph -> if ( -. ph , A , B ) = B )
4		iftrue 3610	. . . 4 |- ( ph -> if ( ph , B , A ) = B )
5	3, 4	eqtr4d 2267	. . 3 |- ( ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
6		iftrue 3610	. . . 4 |- ( -. ph -> if ( -. ph , A , B ) = A )
7		iffalse 3613	. . . 4 |- ( -. ph -> if ( ph , B , A ) = A )
8	6, 7	eqtr4d 2267	. . 3 |- ( -. ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
9	5, 8	jaoi 723	. 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 715  DECID wdc 841   = wceq 1397   ifcif 3605

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606

This theorem is referenced by:  lgsneg  15752  lgsdilem  15755