Theorem ifnot 4133

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

Assertion

Ref	Expression
ifnot	|- if ( -. ph , A , B ) = if ( ph , B , A )

Proof of Theorem ifnot

Step	Hyp	Ref	Expression
1		notnot 136	. . . 4 |- ( ph -> -. -. ph )
2	1	iffalsed 4097	. . 3 |- ( ph -> if ( -. ph , A , B ) = B )
3		iftrue 4092	. . 3 |- ( ph -> if ( ph , B , A ) = B )
4	2, 3	eqtr4d 2659	. 2 |- ( ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
5		iftrue 4092	. . 3 |- ( -. ph -> if ( -. ph , A , B ) = A )
6		iffalse 4095	. . 3 |- ( -. ph -> if ( ph , B , A ) = A )
7	5, 6	eqtr4d 2659	. 2 |- ( -. ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
8	4, 7	pm2.61i 176	1 |- if ( -. ph , A , B ) = if ( ph , B , A )

Syntax hints:   -. wn 3   = wceq 1483   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  suppsnop  7309  2resupmax  12019  sadadd2lem2  15172  maducoeval2  20446  tmsxpsval2  22344  itg2uba  23510  lgsneg  25046  lgsdilem  25049  sgnneg  30602  bj-xpimasn  32942  itgaddnclem2  33469  ftc1anclem5  33489