Theorem dedth4v 4145

Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4143. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Hypotheses

Ref	Expression
dedth4v.1	|- ( A = if ( ph , A , R ) -> ( ps <-> ch ) )
dedth4v.2	|- ( B = if ( ph , B , S ) -> ( ch <-> th ) )
dedth4v.3	|- ( C = if ( ph , C , T ) -> ( th <-> ta ) )
dedth4v.4	|- ( D = if ( ph , D , U ) -> ( ta <-> et ) )
dedth4v.5	|- et

Assertion

Ref	Expression
dedth4v	|- ( ph -> ps )

Proof of Theorem dedth4v

Step	Hyp	Ref	Expression
1		dedth4v.1	. . . 4 |- ( A = if ( ph , A , R ) -> ( ps <-> ch ) )
2		dedth4v.2	. . . 4 |- ( B = if ( ph , B , S ) -> ( ch <-> th ) )
3		dedth4v.3	. . . 4 |- ( C = if ( ph , C , T ) -> ( th <-> ta ) )
4		dedth4v.4	. . . 4 |- ( D = if ( ph , D , U ) -> ( ta <-> et ) )
5		dedth4v.5	. . . 4 |- et
6	1, 2, 3, 4, 5	dedth4h 4142	. . 3 |- ( ( ( ph /\ ph ) /\ ( ph /\ ph ) ) -> ps )
7	6	anidms 677	. 2 |- ( ( ph /\ ph ) -> ps )
8	7	anidms 677	1 |- ( ph -> ps )

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: (None)