Theorem dedth4v 3709

Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3707. (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 3706	. . 3 |- (((ph /\ ph) /\ (ph /\ ph)) -> ps)
7	6	anidms 626	. 2 |- ((ph /\ ph) -> ps)
8	7	anidms 626	1 |- (ph -> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642  ifcif 3662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663

This theorem is referenced by: (None)