Theorem elimhyp2v 4147

Description: Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)

Hypotheses

Ref	Expression
elimhyp2v.1	|- ( A = if ( ph , A , C ) -> ( ph <-> ch ) )
elimhyp2v.2	|- ( B = if ( ph , B , D ) -> ( ch <-> th ) )
elimhyp2v.3	|- ( C = if ( ph , A , C ) -> ( ta <-> et ) )
elimhyp2v.4	|- ( D = if ( ph , B , D ) -> ( et <-> th ) )
elimhyp2v.5	|- ta

Assertion

Ref	Expression
elimhyp2v	|- th

Proof of Theorem elimhyp2v

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , A , C ) = A )
2	1	eqcomd 2628	. . . . 5 |- ( ph -> A = if ( ph , A , C ) )
3		elimhyp2v.1	. . . . 5 |- ( A = if ( ph , A , C ) -> ( ph <-> ch ) )
4	2, 3	syl 17	. . . 4 |- ( ph -> ( ph <-> ch ) )
5		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , B , D ) = B )
6	5	eqcomd 2628	. . . . 5 |- ( ph -> B = if ( ph , B , D ) )
7		elimhyp2v.2	. . . . 5 |- ( B = if ( ph , B , D ) -> ( ch <-> th ) )
8	6, 7	syl 17	. . . 4 |- ( ph -> ( ch <-> th ) )
9	4, 8	bitrd 268	. . 3 |- ( ph -> ( ph <-> th ) )
10	9	ibi 256	. 2 |- ( ph -> th )
11		elimhyp2v.5	. . 3 |- ta
12		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , A , C ) = C )
13	12	eqcomd 2628	. . . . 5 |- ( -. ph -> C = if ( ph , A , C ) )
14		elimhyp2v.3	. . . . 5 |- ( C = if ( ph , A , C ) -> ( ta <-> et ) )
15	13, 14	syl 17	. . . 4 |- ( -. ph -> ( ta <-> et ) )
16		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , B , D ) = D )
17	16	eqcomd 2628	. . . . 5 |- ( -. ph -> D = if ( ph , B , D ) )
18		elimhyp2v.4	. . . . 5 |- ( D = if ( ph , B , D ) -> ( et <-> th ) )
19	17, 18	syl 17	. . . 4 |- ( -. ph -> ( et <-> th ) )
20	15, 19	bitrd 268	. . 3 |- ( -. ph -> ( ta <-> th ) )
21	11, 20	mpbii 223	. 2 |- ( -. ph -> th )
22	10, 21	pm2.61i 176	1 |- th

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:  omlsi  28263