Theorem syl3anr2 1419

Description: A syllogism inference. (Contributed by NM, 1-Aug-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2022.)

Hypotheses

Ref	Expression
syl3anr2.1	⊢ (𝜑 → 𝜃)
syl3anr2.2	⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)

Assertion

Ref	Expression
syl3anr2	⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂)

Proof of Theorem syl3anr2

Step	Hyp	Ref	Expression
1		syl3anr2.1	. . 3 ⊢ (𝜑 → 𝜃)
2	1	3anim2i 1154	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜏))
3		syl3anr2.2	. 2 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜃 ∧ 𝜏)) → 𝜂)
4	2, 3	sylan2 593	1 ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑 ∧ 𝜏)) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 207  df-an 396  df-3an 1089

This theorem is referenced by:  mulgsubdir  19132  dipassr2  30866