Theorem ifor 3008

Description: The possible values of a conditional operator. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem ifor

Step	Hyp	Ref	Expression
1		iftrue 2989	. . . 4 |- (ph -> if(ph, A, B) = A)
2	1	con3i 114	. . 3 |- (-. if(ph, A, B) = A -> -. ph)
3		iffalse 2991	. . 3 |- (-. ph -> if(ph, A, B) = B)
4	2, 3	syl 12	. 2 |- (-. if(ph, A, B) = A -> if(ph, A, B) = B)
5	4	orri 248	1 |- (if(ph, A, B) = A \/ if(ph, A, B) = B)

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

This theorem is referenced by:  ifpr 3077

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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