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  880  reu6  3673  axprlem3OLD  5364  ordelord  6337  f1dmex  7901  soseq  8100  omeulem2  8509  2pwuninel  9061  isumrpcl  15767  kqfvima  23673  caubl  25253  nbupgr  29401  nbumgrvtx  29403  umgr2adedgspth  30005  btwnconn1lem12  36286  omabs2  43763  sbcim2g  44968  ee21an  45161