Theorem dedth3v 4144

Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4143. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Hypotheses

Ref	Expression
dedth3v.1	|- ( A = if ( ph , A , D ) -> ( ps <-> ch ) )
dedth3v.2	|- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
dedth3v.3	|- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
dedth3v.4	|- ta

Assertion

Ref	Expression
dedth3v	|- ( ph -> ps )

Proof of Theorem dedth3v

Step	Hyp	Ref	Expression
1		dedth3v.1	. . . 4 |- ( A = if ( ph , A , D ) -> ( ps <-> ch ) )
2		dedth3v.2	. . . 4 |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )
3		dedth3v.3	. . . 4 |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )
4		dedth3v.4	. . . 4 |- ta
5	1, 2, 3, 4	dedth3h 4141	. . 3 |- ( ( ph /\ ph /\ ph ) -> ps )
6	5	3anidm12 1383	. 2 |- ( ( ph /\ ph ) -> ps )
7	6	anidms 677	1 |- ( ph -> ps )

Syntax hints:   -> 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-3an 1039  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