Theorem mpisyl 1335

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

Hypotheses

Ref	Expression
mpisyl.1	⊢ (𝜑 → 𝜓)
mpisyl.2	⊢ 𝜒
mpisyl.3	⊢ (𝜓 → (𝜒 → 𝜃))

Assertion

Ref	Expression
mpisyl	⊢ (𝜑 → 𝜃)

Proof of Theorem mpisyl

Step	Hyp	Ref	Expression
1		mpisyl.1	. 2 ⊢ (𝜑 → 𝜓)
2		mpisyl.2	. . 3 ⊢ 𝜒
3		mpisyl.3	. . 3 ⊢ (𝜓 → (𝜒 → 𝜃))
4	2, 3	mpi 15	. 2 ⊢ (𝜓 → 𝜃)
5	1, 4	syl 14	1 ⊢ (𝜑 → 𝜃)

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  2592  reusv1  4190  fliftcnv  5435  fliftfun  5436  tfrlemibfn  5942  ordiso  6358