Theorem ifswap 2386

Description: Negating the first argument swaps the last two arguments of a conditional operator.

Assertion

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

Proof of Theorem ifswap

Step	Hyp	Ref	Expression
1		negb 86	. . . 4 |- (ph -> -. -. ph)
2		iffalse 2371	. . . 4 |- (-. -. ph -> if(-. ph, A, B) = B)
3	1, 2	syl 10	. . 3 |- (ph -> if(-. ph, A, B) = B)
4		iftrue 2370	. . 3 |- (ph -> if(ph, B, A) = B)
5	3, 4	eqtr4d 1513	. 2 |- (ph -> if(-. ph, A, B) = if(ph, B, A))
6		iftrue 2370	. . 3 |- (-. ph -> if(-. ph, A, B) = A)
7		iffalse 2371	. . 3 |- (-. ph -> if(ph, B, A) = A)
8	6, 7	eqtr4d 1513	. 2 |- (-. ph -> if(-. ph, A, B) = if(ph, B, A))
9	5, 8	pm2.61i 126	1 |- if(-. ph, A, B) = if(ph, B, A)

Syntax hints:  -. wn 2   = wceq 958  ifcif 2365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-if 2366

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