Theorem syl3anl2 1415

Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2022.)

Hypotheses

Ref	Expression
syl3anl2.1	⊢ (𝜑 → 𝜒)
syl3anl2.2	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl3anl2	⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl2

Step	Hyp	Ref	Expression
1		syl3anl2.1	. . 3 ⊢ (𝜑 → 𝜒)
2	1	3anim2i 1155	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃))
3		syl3anl2.2	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
4	2, 3	sylan 583	1 ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089

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 210  df-an 400  df-3an 1091

This theorem is referenced by:  chfacfscmulcl  21708  chfacfscmulgsum  21711  chfacfpmmulcl  21712  chfacfpmmulgsum  21715  cpmadumatpolylem1  21732  cpmadumatpolylem2  21733  cpmadumatpoly  21734  chcoeffeqlem  21736  2atlt  37139