Theorem ifswap 3010

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		notnot1 102	. . . 4 |- (ph -> -. -. ph)
2		iffalse 2991	. . . 4 |- (-. -. ph -> if(-. ph, A, B) = B)
3	1, 2	syl 12	. . 3 |- (ph -> if(-. ph, A, B) = B)
4		iftrue 2989	. . 3 |- (ph -> if(ph, B, A) = B)
5	3, 4	eqtr4d 1928	. 2 |- (ph -> if(-. ph, A, B) = if(ph, B, A))
6		iftrue 2989	. . 3 |- (-. ph -> if(-. ph, A, B) = A)
7		iffalse 2991	. . 3 |- (-. ph -> if(ph, B, A) = A)
8	6, 7	eqtr4d 1928	. 2 |- (-. ph -> if(-. ph, A, B) = if(ph, B, A))
9	5, 8	pm2.61i 140	1 |- if(-. ph, A, B) = if(ph, B, A)

Syntax hints:  -. wn 2   = wceq 1298  ifcif 2982

This theorem is referenced by:  txmet 15925

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-if 2983

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