Theorem dedth3h 3706

Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3705. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
dedth3h.1	|- (A = if(ph, A, D) -> (th <-> ta))
dedth3h.2	|- (B = if(ps, B, R) -> (ta <-> et))
dedth3h.3	|- (C = if(ch, C, S) -> (et <-> ze))
dedth3h.4	|- ze

Assertion

Ref	Expression
dedth3h	|- ((ph /\ ps /\ ch) -> th)

Proof of Theorem dedth3h

Step	Hyp	Ref	Expression
1		dedth3h.1	. . . 4 |- (A = if(ph, A, D) -> (th <-> ta))
2	1	imbi2d 307	. . 3 |- (A = if(ph, A, D) -> (((ps /\ ch) -> th) <-> ((ps /\ ch) -> ta)))
3		dedth3h.2	. . . 4 |- (B = if(ps, B, R) -> (ta <-> et))
4		dedth3h.3	. . . 4 |- (C = if(ch, C, S) -> (et <-> ze))
5		dedth3h.4	. . . 4 |- ze
6	3, 4, 5	dedth2h 3705	. . 3 |- ((ps /\ ch) -> ta)
7	2, 6	dedth 3704	. 2 |- (ph -> ((ps /\ ch) -> th))
8	7	3impib 1149	1 |- ((ph /\ ps /\ ch) -> th)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642  ifcif 3663

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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by:  dedth3v  3709