Theorem bj-syl66ib 37424

Description: A mixed syllogism inference derived from imbitrdi 254. Shortens bj-dvelimdv1 37764, alexsubALTlem4 24369 (4821>4812), supsrlem 11196 (2868>2863). (Contributed by BJ, 20-Oct-2021.)

Hypotheses

Ref	Expression
bj-syl66ib.1	⊢ (𝜑 → (𝜓 → 𝜃))
bj-syl66ib.2	⊢ (𝜃 → 𝜏)
bj-syl66ib.3	⊢ (𝜏 ↔ 𝜒)

Assertion

Ref	Expression
bj-syl66ib	⊢ (𝜑 → (𝜓 → 𝜒))

Proof of Theorem bj-syl66ib

Step	Hyp	Ref	Expression
1		bj-syl66ib.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜃))
2		bj-syl66ib.2	. . 3 ⊢ (𝜃 → 𝜏)
3	1, 2	syl6 36	. 2 ⊢ (𝜑 → (𝜓 → 𝜏))
4		bj-syl66ib.3	. 2 ⊢ (𝜏 ↔ 𝜒)
5	3, 4	imbitrdi 254	1 ⊢ (𝜑 → (𝜓 → 𝜒))

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:  bj-dvelimdv1  37764