Theorem dedt 767

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

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 764	. 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 194	. 2 |- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> th)
5	1, 4	syl 10	1 |- (ch -> th)

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

This theorem is referenced by:  con3th 768

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