Theorem syl6mpi 67

Description: A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)

Hypotheses

Ref	Expression
syl6mpi.1	|- ( ph -> ( ps -> ch ) )
syl6mpi.2	|- th
syl6mpi.3	|- ( ch -> ( th -> ta ) )

Assertion

Ref	Expression
syl6mpi	|- ( ph -> ( ps -> ta ) )

Proof of Theorem syl6mpi

Step	Hyp	Ref	Expression
1		syl6mpi.1	. 2 |- ( ph -> ( ps -> ch ) )
2		syl6mpi.2	. . 3 |- th
3		syl6mpi.3	. . 3 |- ( ch -> ( th -> ta ) )
4	2, 3	mpi 20	. 2 |- ( ch -> ta )
5	1, 4	syl6 35	1 |- ( ph -> ( ps -> 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:  19.8a  2052  suceloni  7013  bndrank  8704  ac10ct  8857  1re  10039  uspgrn2crct  26700  tratrb  38746  ee20an  38956