Theorem dedth4h 4142

Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4140. (Contributed by NM, 16-May-1999.)

Hypotheses

Ref	Expression
dedth4h.1	|- ( A = if ( ph , A , R ) -> ( ta <-> et ) )
dedth4h.2	|- ( B = if ( ps , B , S ) -> ( et <-> ze ) )
dedth4h.3	|- ( C = if ( ch , C , F ) -> ( ze <-> si ) )
dedth4h.4	|- ( D = if ( th , D , G ) -> ( si <-> rh ) )
dedth4h.5	|- rh

Assertion

Ref	Expression
dedth4h	|- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta )

Proof of Theorem dedth4h

Step	Hyp	Ref	Expression
1		dedth4h.1	. . . 4 |- ( A = if ( ph , A , R ) -> ( ta <-> et ) )
2	1	imbi2d 330	. . 3 |- ( A = if ( ph , A , R ) -> ( ( ( ch /\ th ) -> ta ) <-> ( ( ch /\ th ) -> et ) ) )
3		dedth4h.2	. . . 4 |- ( B = if ( ps , B , S ) -> ( et <-> ze ) )
4	3	imbi2d 330	. . 3 |- ( B = if ( ps , B , S ) -> ( ( ( ch /\ th ) -> et ) <-> ( ( ch /\ th ) -> ze ) ) )
5		dedth4h.3	. . . 4 |- ( C = if ( ch , C , F ) -> ( ze <-> si ) )
6		dedth4h.4	. . . 4 |- ( D = if ( th , D , G ) -> ( si <-> rh ) )
7		dedth4h.5	. . . 4 |- rh
8	5, 6, 7	dedth2h 4140	. . 3 |- ( ( ch /\ th ) -> ze )
9	2, 4, 8	dedth2h 4140	. 2 |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) )
10	9	imp 445	1 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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:  dedth4v  4145  fprg  6422  omopth  7738  nn0opth2  13059  ax5seglem8  25816  hvsubsub4  27917  norm3lemt  28009  eigorth  28697