Theorem syl7bi 221

Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
syl7bi.1	⊢ (φ ↔ ψ)
syl7bi.2	⊢ (χ → (θ → (ψ → τ)))

Assertion

Ref	Expression
syl7bi	⊢ (χ → (θ → (φ → τ)))

Proof of Theorem syl7bi

Step	Hyp	Ref	Expression
1		syl7bi.1	. . 3 ⊢ (φ ↔ ψ)
2	1	biimpi 186	. 2 ⊢ (φ → ψ)
3		syl7bi.2	. 2 ⊢ (χ → (θ → (ψ → τ)))
4	2, 3	syl7 63	1 ⊢ (χ → (θ → (φ → τ)))

Syntax hints:   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  rspct  2949