Theorem ifswap 2353

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 2338	. . . 4 |- (-. -. ph -> if(-. ph, A, B) = B)
3	1, 2	syl 10	. . 3 |- (ph -> if(-. ph, A, B) = B)
4		iftrue 2337	. . 3 |- (ph -> if(ph, B, A) = B)
5	3, 4	eqtr4d 1486	. 2 |- (ph -> if(-. ph, A, B) = if(ph, B, A))
6		iftrue 2337	. . 3 |- (-. ph -> if(-. ph, A, B) = A)
7		iffalse 2338	. . 3 |- (-. ph -> if(ph, B, A) = A)
8	6, 7	eqtr4d 1486	. 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 1099  ifcif 2332

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-if 2333

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