Theorem syl8ib 259

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 219	. 2 ⊢ (𝜃 → 𝜏)
4	1, 3	syl8 77	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

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

This proof depends on definitions:  df-bi 210

This theorem is used by:  en3lplem2  9607  axdc4lem  10526  bj-nexdh  37465