Theorem syl8 65

Description: A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Hypotheses

Ref	Expression
syl8.1	⊢ (φ → (ψ → (χ → θ)))
syl8.2	⊢ (θ → τ)

Assertion

Ref	Expression
syl8	⊢ (φ → (ψ → (χ → τ)))

Proof of Theorem syl8

Step	Hyp	Ref	Expression
1		syl8.1	. 2 ⊢ (φ → (ψ → (χ → θ)))
2		syl8.2	. . 3 ⊢ (θ → τ)
3	2	a1i 10	. 2 ⊢ (φ → (θ → τ))
4	1, 3	syl6d 64	1 ⊢ (φ → (ψ → (χ → 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:  com45  83  syl8ib  222  imp5a  581  3exp  1150  nclenn  6250