Theorem elimh 922

Description: Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)

Hypotheses

Ref	Expression
elimh.1	|- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (ch <-> ta))
elimh.2	|- ((ps <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta))
elimh.3	|- th

Assertion

Ref	Expression
elimh	|- ta

Proof of Theorem elimh

Step	Hyp	Ref	Expression
1		dedlema 920	. . . 4 |- (ch -> (ph <-> ((ph /\ ch) \/ (ps /\ -. ch))))
2		elimh.1	. . . 4 |- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (ch <-> ta))
3	1, 2	syl 15	. . 3 |- (ch -> (ch <-> ta))
4	3	ibi 232	. 2 |- (ch -> ta)
5		elimh.3	. . 3 |- th
6		dedlemb 921	. . . 4 |- (-. ch -> (ps <-> ((ph /\ ch) \/ (ps /\ -. ch))))
7		elimh.2	. . . 4 |- ((ps <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta))
8	6, 7	syl 15	. . 3 |- (-. ch -> (th <-> ta))
9	5, 8	mpbii 202	. 2 |- (-. ch -> ta)
10	4, 9	pm2.61i 156	1 |- ta

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by:  con3th  924