Theorem mpisyl 1446

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-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  ceqsex  2777  reusv1  4460  fliftcnv  5798  fliftfun  5799  tfrlemibfn  6331  tfr1onlembfn  6347  tfrcllembfn  6360  cnvct  6811  ordiso  7037  exmidomni  7142  djudoml  7220  djudomr  7221  uzsinds  10444  fimaxq  10809  ltoddhalfle  11900  phicl2  12216  strsetsid  12497  txdis1cn  13817  xmeter  13975  subctctexmid  14789