Theorem keephyp2v 4153

Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4139). (Contributed by NM, 16-Apr-2005.)

Hypotheses

Ref	Expression
keephyp2v.1	|- ( A = if ( ph , A , C ) -> ( ps <-> ch ) )
keephyp2v.2	|- ( B = if ( ph , B , D ) -> ( ch <-> th ) )
keephyp2v.3	|- ( C = if ( ph , A , C ) -> ( ta <-> et ) )
keephyp2v.4	|- ( D = if ( ph , B , D ) -> ( et <-> th ) )
keephyp2v.5	|- ps
keephyp2v.6	|- ta

Assertion

Ref	Expression
keephyp2v	|- th

Proof of Theorem keephyp2v

Step	Hyp	Ref	Expression
1		keephyp2v.5	. . 3 |- ps
2		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , A , C ) = A )
3	2	eqcomd 2628	. . . . 5 |- ( ph -> A = if ( ph , A , C ) )
4		keephyp2v.1	. . . . 5 |- ( A = if ( ph , A , C ) -> ( ps <-> ch ) )
5	3, 4	syl 17	. . . 4 |- ( ph -> ( ps <-> ch ) )
6		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , B , D ) = B )
7	6	eqcomd 2628	. . . . 5 |- ( ph -> B = if ( ph , B , D ) )
8		keephyp2v.2	. . . . 5 |- ( B = if ( ph , B , D ) -> ( ch <-> th ) )
9	7, 8	syl 17	. . . 4 |- ( ph -> ( ch <-> th ) )
10	5, 9	bitrd 268	. . 3 |- ( ph -> ( ps <-> th ) )
11	1, 10	mpbii 223	. 2 |- ( ph -> th )
12		keephyp2v.6	. . 3 |- ta
13		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , A , C ) = C )
14	13	eqcomd 2628	. . . . 5 |- ( -. ph -> C = if ( ph , A , C ) )
15		keephyp2v.3	. . . . 5 |- ( C = if ( ph , A , C ) -> ( ta <-> et ) )
16	14, 15	syl 17	. . . 4 |- ( -. ph -> ( ta <-> et ) )
17		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , B , D ) = D )
18	17	eqcomd 2628	. . . . 5 |- ( -. ph -> D = if ( ph , B , D ) )
19		keephyp2v.4	. . . . 5 |- ( D = if ( ph , B , D ) -> ( et <-> th ) )
20	18, 19	syl 17	. . . 4 |- ( -. ph -> ( et <-> th ) )
21	16, 20	bitrd 268	. . 3 |- ( -. ph -> ( ta <-> th ) )
22	12, 21	mpbii 223	. 2 |- ( -. ph -> th )
23	11, 22	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: (None)