Theorem syl8ib 166

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 120	. 2 ⊢ (𝜃 → 𝜏)
4	1, 3	syl8 71	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm3.2an3  1176  necon4bddc  2418  necon4abiddc  2420  necon4bbiddc  2421  necon4biddc  2422