Theorem mpisyl 1380

Description: A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)

Hypotheses

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

Assertion

Ref	Expression
mpisyl	|- ( ph -> th )

Proof of Theorem mpisyl

Step	Hyp	Ref	Expression
1		mpisyl.1	. 2 |- ( ph -> ps )
2		mpisyl.2	. . 3 |- ch
3		mpisyl.3	. . 3 |- ( ps -> ( ch -> th ) )
4	2, 3	mpi 15	. 2 |- ( ps -> th )
5	1, 4	syl 14	1 |- ( ph -> th )

Syntax hints:   -> wi 4

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

This theorem is referenced by:  ceqsex  2657  reusv1  4278  fliftcnv  5566  fliftfun  5567  tfrlemibfn  6085  tfr1onlembfn  6101  tfrcllembfn  6114  cnvct  6516  ordiso  6719  exmidomni  6788  uzsinds  9836  fimaxq  10223  ltoddhalfle  11158  phicl2  11455