Theorem syl8ib 222

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.)

Hypotheses

Ref	Expression
syl8ib.1	⊢ (φ → (ψ → (χ → θ)))
syl8ib.2	⊢ (θ ↔ τ)

Assertion

Ref	Expression
syl8ib	⊢ (φ → (ψ → (χ → τ)))

Proof of Theorem syl8ib

Step	Hyp	Ref	Expression
1		syl8ib.1	. 2 ⊢ (φ → (ψ → (χ → θ)))
2		syl8ib.2	. . 3 ⊢ (θ ↔ τ)
3	2	biimpi 186	. 2 ⊢ (θ → τ)
4	1, 3	syl8 65	1 ⊢ (φ → (ψ → (χ → τ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  pm3.2an3  1131  ncfinraise  4482