Theorem 3syld 56

Description: Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)

Hypotheses

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

Assertion

Ref	Expression
3syld	|- ( ph -> ( ps -> ta ) )

Proof of Theorem 3syld

Step	Hyp	Ref	Expression
1		3syld.1	. . 3 |- ( ph -> ( ps -> ch ) )
2		3syld.2	. . 3 |- ( ph -> ( ch -> th ) )
3	1, 2	syld 44	. 2 |- ( ph -> ( ps -> th ) )
4		3syld.3	. 2 |- ( ph -> ( th -> ta ) )
5	3, 4	syld 44	1 |- ( ph -> ( ps -> ta ) )

Syntax hints:   -> wi 4

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

This theorem is referenced by:  apreap  7687  msqge0  7716  cju  8038  facavg  9673  mulcn2  10151  coprm  10523  rpexp  10532