Theorem syl10 79

Description: A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)

Hypotheses

Ref	Expression
syl10.1	|- ( ph -> ( ps -> ch ) )
syl10.2	|- ( ph -> ( ps -> ( th -> ta ) ) )
syl10.3	|- ( ch -> ( ta -> et ) )

Assertion

Ref	Expression
syl10	|- ( ph -> ( ps -> ( th -> et ) ) )

Proof of Theorem syl10

Step	Hyp	Ref	Expression
1		syl10.2	. 2 |- ( ph -> ( ps -> ( th -> ta ) ) )
2		syl10.1	. . 3 |- ( ph -> ( ps -> ch ) )
3		syl10.3	. . 3 |- ( ch -> ( ta -> et ) )
4	2, 3	syl6 35	. 2 |- ( ph -> ( ps -> ( ta -> et ) ) )
5	1, 4	syldd 72	1 |- ( ph -> ( ps -> ( th -> et ) ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  tz7.49  7540  rspsbc2  38744  tratrb  38746