Theorem iforOLD 3009

Description: The possible values of a conditional operator.

Assertion

Ref	Expression
iforOLD	|- (if(ph, A, B) = A \/ if(ph, A, B) = B)

Proof of Theorem iforOLD

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- if(ph, A, B) = if(ph, A, B)
2		eqif 3004	. . 3 |- (if(ph, A, B) = if(ph, A, B) <-> ((ph /\ if(ph, A, B) = A) \/ (-. ph /\ if(ph, A, B) = B)))
3	1, 2	mpbi 206	. 2 |- ((ph /\ if(ph, A, B) = A) \/ (-. ph /\ if(ph, A, B) = B))
4		simpr 350	. . 3 |- ((ph /\ if(ph, A, B) = A) -> if(ph, A, B) = A)
5		simpr 350	. . 3 |- ((-. ph /\ if(ph, A, B) = B) -> if(ph, A, B) = B)
6	4, 5	orim12i 363	. 2 |- (((ph /\ if(ph, A, B) = A) \/ (-. ph /\ if(ph, A, B) = B)) -> (if(ph, A, B) = A \/ if(ph, A, B) = B))
7	3, 6	ax-mp 7	1 |- (if(ph, A, B) = A \/ if(ph, A, B) = B)

Syntax hints:  -. wn 2   \/ wo 239   /\ wa 240   = wceq 1298  ifcif 2982

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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