Theorem elimhyp4v 4149

Description: Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4139). (Contributed by NM, 16-Apr-2005.)

Hypotheses

Ref	Expression
elimhyp4v.1	|- ( A = if ( ph , A , D ) -> ( ph <-> ch ) )
elimhyp4v.2	|- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
elimhyp4v.3	|- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
elimhyp4v.4	|- ( F = if ( ph , F , G ) -> ( ta <-> ps ) )
elimhyp4v.5	|- ( D = if ( ph , A , D ) -> ( et <-> ze ) )
elimhyp4v.6	|- ( R = if ( ph , B , R ) -> ( ze <-> si ) )
elimhyp4v.7	|- ( S = if ( ph , C , S ) -> ( si <-> rh ) )
elimhyp4v.8	|- ( G = if ( ph , F , G ) -> ( rh <-> ps ) )
elimhyp4v.9	|- et

Assertion

Ref	Expression
elimhyp4v	|- ps

Proof of Theorem elimhyp4v

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . . . . 7 |- ( ph -> if ( ph , A , D ) = A )
2	1	eqcomd 2628	. . . . . 6 |- ( ph -> A = if ( ph , A , D ) )
3		elimhyp4v.1	. . . . . 6 |- ( A = if ( ph , A , D ) -> ( ph <-> ch ) )
4	2, 3	syl 17	. . . . 5 |- ( ph -> ( ph <-> ch ) )
5		iftrue 4092	. . . . . . 7 |- ( ph -> if ( ph , B , R ) = B )
6	5	eqcomd 2628	. . . . . 6 |- ( ph -> B = if ( ph , B , R ) )
7		elimhyp4v.2	. . . . . 6 |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
8	6, 7	syl 17	. . . . 5 |- ( ph -> ( ch <-> th ) )
9	4, 8	bitrd 268	. . . 4 |- ( ph -> ( ph <-> th ) )
10		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , C , S ) = C )
11	10	eqcomd 2628	. . . . 5 |- ( ph -> C = if ( ph , C , S ) )
12		elimhyp4v.3	. . . . 5 |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
13	11, 12	syl 17	. . . 4 |- ( ph -> ( th <-> ta ) )
14		iftrue 4092	. . . . . 6 |- ( ph -> if ( ph , F , G ) = F )
15	14	eqcomd 2628	. . . . 5 |- ( ph -> F = if ( ph , F , G ) )
16		elimhyp4v.4	. . . . 5 |- ( F = if ( ph , F , G ) -> ( ta <-> ps ) )
17	15, 16	syl 17	. . . 4 |- ( ph -> ( ta <-> ps ) )
18	9, 13, 17	3bitrd 294	. . 3 |- ( ph -> ( ph <-> ps ) )
19	18	ibi 256	. 2 |- ( ph -> ps )
20		elimhyp4v.9	. . 3 |- et
21		iffalse 4095	. . . . . . 7 |- ( -. ph -> if ( ph , A , D ) = D )
22	21	eqcomd 2628	. . . . . 6 |- ( -. ph -> D = if ( ph , A , D ) )
23		elimhyp4v.5	. . . . . 6 |- ( D = if ( ph , A , D ) -> ( et <-> ze ) )
24	22, 23	syl 17	. . . . 5 |- ( -. ph -> ( et <-> ze ) )
25		iffalse 4095	. . . . . . 7 |- ( -. ph -> if ( ph , B , R ) = R )
26	25	eqcomd 2628	. . . . . 6 |- ( -. ph -> R = if ( ph , B , R ) )
27		elimhyp4v.6	. . . . . 6 |- ( R = if ( ph , B , R ) -> ( ze <-> si ) )
28	26, 27	syl 17	. . . . 5 |- ( -. ph -> ( ze <-> si ) )
29	24, 28	bitrd 268	. . . 4 |- ( -. ph -> ( et <-> si ) )
30		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , C , S ) = S )
31	30	eqcomd 2628	. . . . 5 |- ( -. ph -> S = if ( ph , C , S ) )
32		elimhyp4v.7	. . . . 5 |- ( S = if ( ph , C , S ) -> ( si <-> rh ) )
33	31, 32	syl 17	. . . 4 |- ( -. ph -> ( si <-> rh ) )
34		iffalse 4095	. . . . . 6 |- ( -. ph -> if ( ph , F , G ) = G )
35	34	eqcomd 2628	. . . . 5 |- ( -. ph -> G = if ( ph , F , G ) )
36		elimhyp4v.8	. . . . 5 |- ( G = if ( ph , F , G ) -> ( rh <-> ps ) )
37	35, 36	syl 17	. . . 4 |- ( -. ph -> ( rh <-> ps ) )
38	29, 33, 37	3bitrd 294	. . 3 |- ( -. ph -> ( et <-> ps ) )
39	20, 38	mpbii 223	. 2 |- ( -. ph -> ps )
40	19, 39	pm2.61i 176	1 |- ps

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)