Theorem dedt 572

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 569	. 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 169	. 2 |- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> th)
5	1, 4	syl 12	1 |- (ch -> th)

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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