Theorem ifbi 2371

Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)

Assertion

Ref	Expression
ifbi	|- ((ph <-> ps) -> if(ph, A, B) = if(ps, A, B))

Proof of Theorem ifbi

Step	Hyp	Ref	Expression
1		dfbi3 670	. 2 |- ((ph <-> ps) <-> ((ph /\ ps) \/ (-. ph /\ -. ps)))
2		iftrue 2366	. . . 4 |- (ph -> if(ph, A, B) = A)
3		iftrue 2366	. . . . 5 |- (ps -> if(ps, A, B) = A)
4	3	eqcomd 1480	. . . 4 |- (ps -> A = if(ps, A, B))
5	2, 4	sylan9eq 1527	. . 3 |- ((ph /\ ps) -> if(ph, A, B) = if(ps, A, B))
6		iffalse 2367	. . . 4 |- (-. ph -> if(ph, A, B) = B)
7		iffalse 2367	. . . . 5 |- (-. ps -> if(ps, A, B) = B)
8	7	eqcomd 1480	. . . 4 |- (-. ps -> B = if(ps, A, B))
9	6, 8	sylan9eq 1527	. . 3 |- ((-. ph /\ -. ps) -> if(ph, A, B) = if(ps, A, B))
10	5, 9	jaoi 341	. 2 |- (((ph /\ ps) \/ (-. ph /\ -. ps)) -> if(ph, A, B) = if(ps, A, B))
11	1, 10	sylbi 199	1 |- ((ph <-> ps) -> if(ph, A, B) = if(ps, A, B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956  ifcif 2361

This theorem is referenced by:  ifbid 2372  ruclem15 7524

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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