Theorem dedth4h 3706

Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3704. (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 307	. . 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 307	. . 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 3704	. . 3 |- ((ch /\ th) -> ze)
9	2, 4, 8	dedth2h 3704	. 2 |- ((ph /\ ps) -> ((ch /\ th) -> ta))
10	9	imp 418	1 |- (((ph /\ ps) /\ (ch /\ th)) -> ta)

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  dedth4v  3709