Theorem syl3anl2 1375

Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)

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		syl3anl2.2	. . . 4 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
3	2	ex 450	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜏 → 𝜂))
4	1, 3	syl3an2 1360	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → (𝜏 → 𝜂))
5	4	imp 445	1 ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  syl3anr2  1379  chfacfscmulcl  20662  chfacfscmulgsum  20665  chfacfpmmulcl  20666  chfacfpmmulgsum  20669  cpmadumatpolylem1  20686  cpmadumatpolylem2  20687  cpmadumatpoly  20688  chcoeffeqlem  20690  2atlt  34725