Theorem keephyp3v 4154

Description: Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)

Hypotheses

Ref	Expression
keephyp3v.1	|- ( A = if ( ph , A , D ) -> ( rh <-> ch ) )
keephyp3v.2	|- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
keephyp3v.3	|- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
keephyp3v.4	|- ( D = if ( ph , A , D ) -> ( et <-> ze ) )
keephyp3v.5	|- ( R = if ( ph , B , R ) -> ( ze <-> si ) )
keephyp3v.6	|- ( S = if ( ph , C , S ) -> ( si <-> ta ) )
keephyp3v.7	|- rh
keephyp3v.8	|- et

Assertion

Ref	Expression
keephyp3v	|- ta

Proof of Theorem keephyp3v

Step	Hyp	Ref	Expression
1		keephyp3v.7	. . 3 |- rh
2		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , A , D ) = A )
3	2	eqcomd 2628	. . . . 5 |- ( ph -> A = if ( ph , A , D ) )
4		keephyp3v.1	. . . . 5 |- ( A = if ( ph , A , D ) -> ( rh <-> ch ) )
5	3, 4	syl 17	. . . 4 |- ( ph -> ( rh <-> ch ) )
6		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , B , R ) = B )
7	6	eqcomd 2628	. . . . 5 |- ( ph -> B = if ( ph , B , R ) )
8		keephyp3v.2	. . . . 5 |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
9	7, 8	syl 17	. . . 4 |- ( ph -> ( ch <-> th ) )
10		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , C , S ) = C )
11	10	eqcomd 2628	. . . . 5 |- ( ph -> C = if ( ph , C , S ) )
12		keephyp3v.3	. . . . 5 |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
13	11, 12	syl 17	. . . 4 |- ( ph -> ( th <-> ta ) )
14	5, 9, 13	3bitrd 294	. . 3 |- ( ph -> ( rh <-> ta ) )
15	1, 14	mpbii 223	. 2 |- ( ph -> ta )
16		keephyp3v.8	. . 3 |- et
17		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , A , D ) = D )
18	17	eqcomd 2628	. . . . 5 |- ( -. ph -> D = if ( ph , A , D ) )
19		keephyp3v.4	. . . . 5 |- ( D = if ( ph , A , D ) -> ( et <-> ze ) )
20	18, 19	syl 17	. . . 4 |- ( -. ph -> ( et <-> ze ) )
21		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , B , R ) = R )
22	21	eqcomd 2628	. . . . 5 |- ( -. ph -> R = if ( ph , B , R ) )
23		keephyp3v.5	. . . . 5 |- ( R = if ( ph , B , R ) -> ( ze <-> si ) )
24	22, 23	syl 17	. . . 4 |- ( -. ph -> ( ze <-> si ) )
25		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , C , S ) = S )
26	25	eqcomd 2628	. . . . 5 |- ( -. ph -> S = if ( ph , C , S ) )
27		keephyp3v.6	. . . . 5 |- ( S = if ( ph , C , S ) -> ( si <-> ta ) )
28	26, 27	syl 17	. . . 4 |- ( -. ph -> ( si <-> ta ) )
29	20, 24, 28	3bitrd 294	. . 3 |- ( -. ph -> ( et <-> ta ) )
30	16, 29	mpbii 223	. 2 |- ( -. ph -> ta )
31	15, 30	pm2.61i 176	1 |- ta

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:  sseliALT  4791