Theorem syl6ci 71

Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)

Hypotheses

Ref	Expression
syl6ci.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl6ci.2	⊢ (𝜑 → 𝜃)
syl6ci.3	⊢ (𝜒 → (𝜃 → 𝜏))

Assertion

Ref	Expression
syl6ci	⊢ (𝜑 → (𝜓 → 𝜏))

Proof of Theorem syl6ci

Step	Hyp	Ref	Expression
1		syl6ci.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		syl6ci.2	. . 3 ⊢ (𝜑 → 𝜃)
3	2	a1d 25	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
4		syl6ci.3	. 2 ⊢ (𝜒 → (𝜃 → 𝜏))
5	1, 3, 4	syl6c 70	1 ⊢ (𝜑 → (𝜓 → 𝜏))

Syntax hints:   → wi 4

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

This theorem is referenced by:  mtord  878  reu6  3719  axprlem3  5419  ordelord  6385  f1dmex  7954  soseq  8158  omeulem2  8597  2pwuninel  9148  isumrpcl  15813  kqfvima  23621  caubl  25223  nbupgr  29144  nbumgrvtx  29146  umgr2adedgspth  29746  btwnconn1lem12  35630  omabs2  42684  sbcim2g  43900  ee21an  44094