Theorem elimh 571

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 569	. . . 4 |- (ch -> (ph <-> ((ph /\ ch) \/ (ps /\ -. ch))))
2		elimh.1	. . . 4 |- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (ch <-> ta ))
3	1, 2	syl 12	. . 3 |- (ch -> (ch <-> ta ))
4	3	ibi 449	. 2 |- (ch -> ta )
5		elimh.3	. . 3 |- th
6		dedlemb 570	. . . 4 |- (-. ch -> (ps <-> ((ph /\ ch) \/ (ps /\ -. ch))))
7		elimh.2	. . . 4 |- ((ps <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta ))
8	6, 7	syl 12	. . 3 |- (-. ch -> (th <-> ta ))
9	5, 8	mpbii 168	. 2 |- (-. ch -> ta )
10	4, 9	pm2.61i 110	1 |- ta

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

This theorem is referenced by:  con3th 573

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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