Theorem syl2and 354

Description: A syllogism deduction.

Hypotheses

Ref	Expression
syl2and.1	|- (ph -> ((ps /\ ch) -> th))
syl2and.2	|- (ph -> (ta -> ps))
syl2and.3	|- (ph -> (et -> ch))

Assertion

Ref	Expression
syl2and	|- (ph -> ((ta /\ et) -> th))

Proof of Theorem syl2and

Step	Hyp	Ref	Expression
1		syl2and.1	. . 3 |- (ph -> ((ps /\ ch) -> th))
2		syl2and.3	. . 3 |- (ph -> (et -> ch))
3	1, 2	sylan2d 353	. 2 |- (ph -> ((ps /\ et) -> th))
4		syl2and.2	. 2 |- (ph -> (ta -> ps))
5	3, 4	syland 352	1 |- (ph -> ((ta /\ et) -> th))

Syntax hints:   -> wi 2   /\ wa 196

This theorem is referenced by:  shsvst 5288

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-an 198

metamath.org