Theorem ifor 2381

Description: The possible values of a conditional operator.

Assertion

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

Proof of Theorem ifor

Step	Hyp	Ref	Expression
1		eqid 1475	. . 3 |- if(ph, A, B) = if(ph, A, B)
2		eqif 2377	. . 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 189	. 2 |- ((ph /\ if(ph, A, B) = A) \/ (-. ph /\ if(ph, A, B) = B))
4		pm3.27 323	. . 3 |- ((ph /\ if(ph, A, B) = A) -> if(ph, A, B) = A)
5		pm3.27 323	. . 3 |- ((-. ph /\ if(ph, A, B) = B) -> if(ph, A, B) = B)
6	4, 5	orim12i 336	. 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 222   /\ wa 223   = wceq 956  ifcif 2361

This theorem is referenced by:  ifpr 2427

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2362

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