Theorem syl3anl2 1231

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 423	. . 3 ⊢ ((ψ ∧ χ ∧ θ) → (τ → η))
4	1, 3	syl3an2 1216	. 2 ⊢ ((ψ ∧ φ ∧ θ) → (τ → η))
5	4	imp 418	1 ⊢ (((ψ ∧ φ ∧ θ) ∧ τ) → η)

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934

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 177  df-an 360  df-3an 936

This theorem is referenced by:  syl3anr2  1235