Theorem syl6ci 1495

Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)

Hypotheses

Ref	Expression
syl6ci.1	|- ( ph -> ( ps -> ch ) )
syl6ci.2	|- ( ph -> th )
syl6ci.3	|- ( ch -> ( th -> ta ) )

Assertion

Ref	Expression
syl6ci	|- ( ph -> ( ps -> ta ) )

Proof of Theorem syl6ci

Step	Hyp	Ref	Expression
1		syl6ci.1	. 2 |- ( ph -> ( ps -> ch ) )
2		syl6ci.2	. . 3 |- ( ph -> th )
3	2	a1d 22	. 2 |- ( ph -> ( ps -> th ) )
4		syl6ci.3	. 2 |- ( ch -> ( th -> ta ) )
5	1, 3, 4	syl6c 66	1 |- ( ph -> ( ps -> ta ) )

This proof depends on syntax axioms:   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  ltxrlt  8391  ltnsym  8411  absle  11855  isumrpcl  12261  lealltlt1  16751  lealltlt2  16752