Theorem bj-syl66ib 33890

Description: A mixed syllogism inference derived from syl6ib 253. In addition to bj-dvelimdv1 34176, it can also shorten alexsubALTlem4 22657 (4821>4812), supsrlem 10532 (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 35	. 2 ⊢ (𝜑 → (𝜓 → 𝜏))
4		bj-syl66ib.3	. 2 ⊢ (𝜏 ↔ 𝜒)
5	3, 4	syl6ib 253	1 ⊢ (𝜑 → (𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208

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 209

This theorem is referenced by:  bj-dvelimdv1  34176