Theorem elimdhyp 4151

Description: Version of elimhyp 4146 where the hypothesis is deduced from the final antecedent. See divalg 15126 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 4092	. . . . 5 |- ( ph -> if ( ph , A , B ) = A )
3	2	eqcomd 2628	. . . 4 |- ( ph -> A = if ( ph , A , B ) )
4		elimdhyp.2	. . . 4 |- ( A = if ( ph , A , B ) -> ( ps <-> ch ) )
5	3, 4	syl 17	. . 3 |- ( ph -> ( ps <-> ch ) )
6	1, 5	mpbid 222	. 2 |- ( ph -> ch )
7		elimdhyp.4	. . 3 |- th
8		iffalse 4095	. . . . 5 |- ( -. ph -> if ( ph , A , B ) = B )
9	8	eqcomd 2628	. . . 4 |- ( -. ph -> B = if ( ph , A , B ) )
10		elimdhyp.3	. . . 4 |- ( B = if ( ph , A , B ) -> ( th <-> ch ) )
11	9, 10	syl 17	. . 3 |- ( -. ph -> ( th <-> ch ) )
12	7, 11	mpbii 223	. 2 |- ( -. ph -> ch )
13	6, 12	pm2.61i 176	1 |- ch

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  divalg  15126