Theorem elimdhyp 3715

Description: Version of elimhyp 3710 where the hypothesis is deduced from the final antecedent. See ghomgrplem in set.mm 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 3668	. . . . 5 |- (ph -> if(ph, A, B) = A)
3	2	eqcomd 2358	. . . 4 |- (ph -> A = if(ph, A, B))
4		elimdhyp.2	. . . 4 |- (A = if(ph, A, B) -> (ps <-> ch))
5	3, 4	syl 15	. . 3 |- (ph -> (ps <-> ch))
6	1, 5	mpbid 201	. 2 |- (ph -> ch)
7		elimdhyp.4	. . 3 |- th
8		iffalse 3669	. . . . 5 |- (-. ph -> if(ph, A, B) = B)
9	8	eqcomd 2358	. . . 4 |- (-. ph -> B = if(ph, A, B))
10		elimdhyp.3	. . . 4 |- (B = if(ph, A, B) -> (th <-> ch))
11	9, 10	syl 15	. . 3 |- (-. ph -> (th <-> ch))
12	7, 11	mpbii 202	. 2 |- (-. ph -> ch)
13	6, 12	pm2.61i 156	1 |- ch

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   = wceq 1642  ifcif 3662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663

This theorem is referenced by: (None)