Theorem syldd 67

Description: Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)

Hypotheses

Ref	Expression
syldd.1	|- ( ph -> ( ps -> ( ch -> th ) ) )
syldd.2	|- ( ph -> ( ps -> ( th -> ta ) ) )

Assertion

Ref	Expression
syldd	|- ( ph -> ( ps -> ( ch -> ta ) ) )

Proof of Theorem syldd

Step	Hyp	Ref	Expression
1		syldd.2	. 2 |- ( ph -> ( ps -> ( th -> ta ) ) )
2		syldd.1	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
3		imim2 55	. 2 |- ( ( th -> ta ) -> ( ( ch -> th ) -> ( ch -> ta ) ) )
4	1, 2, 3	syl6c 66	1 |- ( ph -> ( ps -> ( ch -> ta ) ) )

Syntax hints:   -> wi 4

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

This theorem is referenced by:  syl5d  68  syl6d  70  syl10  1428  ordiso2  7012  oddprmdvds  12306