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  3748  axprlem3  5443  ordelord  6417  f1dmex  7997  soseq  8200  omeulem2  8639  2pwuninel  9198  isumrpcl  15891  kqfvima  23759  caubl  25361  nbupgr  29379  nbumgrvtx  29381  umgr2adedgspth  29981  btwnconn1lem12  36062  omabs2  43294  sbcim2g  44509  ee21an  44703