Theorem elimdhyp 2385

Description: Version of elimhyp 2380 where the hypothesis is deduced from the final antecedent. See ghomgrplem 10294 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Hypotheses

Ref	Expression
elimdhyp.1	|- (ph -> ps)
elimdhyp.2	|- (A = if(ph, A, B) -> (ps <-> ch))
elimdhyp.3	|- (B = if(ph, A, B) -> (th <-> ch))
elimdhyp.4	|- th

Assertion

Ref	Expression
elimdhyp	|- ch

Proof of Theorem elimdhyp

Step	Hyp	Ref	Expression
1		elimdhyp.1	. . 3 |- (ph -> ps)
2		iftrue 2356	. . . . 5 |- (ph -> if(ph, A, B) = A)
3	2	eqcomd 1472	. . . 4 |- (ph -> A = if(ph, A, B))
4		elimdhyp.2	. . . 4 |- (A = if(ph, A, B) -> (ps <-> ch))
5	3, 4	syl 10	. . 3 |- (ph -> (ps <-> ch))
6	1, 5	mpbid 195	. 2 |- (ph -> ch)
7		elimdhyp.4	. . 3 |- th
8		iffalse 2357	. . . . 5 |- (-. ph -> if(ph, A, B) = B)
9	8	eqcomd 1472	. . . 4 |- (-. ph -> B = if(ph, A, B))
10		elimdhyp.3	. . . 4 |- (B = if(ph, A, B) -> (th <-> ch))
11	9, 10	syl 10	. . 3 |- (-. ph -> (th <-> ch))
12	7, 11	mpbii 193	. 2 |- (-. ph -> ch)
13	6, 12	pm2.61i 126	1 |- ch

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 953  ifcif 2351

This theorem is referenced by:  ghomgrplem 10294

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-if 2352

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