Theorem elimh 763

Description: Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page.

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 761	. . . 4 |- (ch -> (ph <-> ((ph /\ ch) \/ (ps /\ -. ch))))
2		elimh.1	. . . 4 |- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (ch <-> ta))
3	1, 2	syl 10	. . 3 |- (ch -> (ch <-> ta))
4	3	ibi 591	. 2 |- (ch -> ta)
5		elimh.3	. . 3 |- th
6		dedlemb 762	. . . 4 |- (-. ch -> (ps <-> ((ph /\ ch) \/ (ps /\ -. ch))))
7		elimh.2	. . . 4 |- ((ps <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta))
8	6, 7	syl 10	. . 3 |- (-. ch -> (th <-> ta))
9	5, 8	mpbii 193	. 2 |- (-. ch -> ta)
10	4, 9	pm2.61i 126	1 |- ta

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  con3th 765

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

Copyright terms: Public domain