Theorem dedt 923

Description: The 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
dedt.1	|- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta))
dedt.2	|- ta

Assertion

Ref	Expression
dedt	|- (ch -> th)

Proof of Theorem dedt

Step	Hyp	Ref	Expression
1		dedlema 920	. 2 |- (ch -> (ph <-> ((ph /\ ch) \/ (ps /\ -. ch))))
2		dedt.2	. . 3 |- ta
3		dedt.1	. . 3 |- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta))
4	2, 3	mpbiri 224	. 2 |- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> th)
5	1, 4	syl 15	1 |- (ch -> th)

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