Theorem elimhyp3v 4148

Description: Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)

Hypotheses

Ref	Expression
elimhyp3v.1	|- ( A = if ( ph , A , D ) -> ( ph <-> ch ) )
elimhyp3v.2	|- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
elimhyp3v.3	|- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
elimhyp3v.4	|- ( D = if ( ph , A , D ) -> ( et <-> ze ) )
elimhyp3v.5	|- ( R = if ( ph , B , R ) -> ( ze <-> si ) )
elimhyp3v.6	|- ( S = if ( ph , C , S ) -> ( si <-> ta ) )
elimhyp3v.7	|- et

Assertion

Ref	Expression
elimhyp3v	|- ta

Proof of Theorem elimhyp3v

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , A , D ) = A )
2	1	eqcomd 2628	. . . . 5 |- ( ph -> A = if ( ph , A , D ) )
3		elimhyp3v.1	. . . . 5 |- ( A = if ( ph , A , D ) -> ( ph <-> ch ) )
4	2, 3	syl 17	. . . 4 |- ( ph -> ( ph <-> ch ) )
5		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , B , R ) = B )
6	5	eqcomd 2628	. . . . 5 |- ( ph -> B = if ( ph , B , R ) )
7		elimhyp3v.2	. . . . 5 |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
8	6, 7	syl 17	. . . 4 |- ( ph -> ( ch <-> th ) )
9		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , C , S ) = C )
10	9	eqcomd 2628	. . . . 5 |- ( ph -> C = if ( ph , C , S ) )
11		elimhyp3v.3	. . . . 5 |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
12	10, 11	syl 17	. . . 4 |- ( ph -> ( th <-> ta ) )
13	4, 8, 12	3bitrd 294	. . 3 |- ( ph -> ( ph <-> ta ) )
14	13	ibi 256	. 2 |- ( ph -> ta )
15		elimhyp3v.7	. . 3 |- et
16		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , A , D ) = D )
17	16	eqcomd 2628	. . . . 5 |- ( -. ph -> D = if ( ph , A , D ) )
18		elimhyp3v.4	. . . . 5 |- ( D = if ( ph , A , D ) -> ( et <-> ze ) )
19	17, 18	syl 17	. . . 4 |- ( -. ph -> ( et <-> ze ) )
20		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , B , R ) = R )
21	20	eqcomd 2628	. . . . 5 |- ( -. ph -> R = if ( ph , B , R ) )
22		elimhyp3v.5	. . . . 5 |- ( R = if ( ph , B , R ) -> ( ze <-> si ) )
23	21, 22	syl 17	. . . 4 |- ( -. ph -> ( ze <-> si ) )
24		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , C , S ) = S )
25	24	eqcomd 2628	. . . . 5 |- ( -. ph -> S = if ( ph , C , S ) )
26		elimhyp3v.6	. . . . 5 |- ( S = if ( ph , C , S ) -> ( si <-> ta ) )
27	25, 26	syl 17	. . . 4 |- ( -. ph -> ( si <-> ta ) )
28	19, 23, 27	3bitrd 294	. . 3 |- ( -. ph -> ( et <-> ta ) )
29	15, 28	mpbii 223	. 2 |- ( -. ph -> ta )
30	14, 29	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