Theorem syl3anl1 1230

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

Hypotheses

Ref	Expression
syl3anl1.1	⊢ (φ → ψ)
syl3anl1.2	⊢ (((ψ ∧ χ ∧ θ) ∧ τ) → η)

Assertion

Ref	Expression
syl3anl1	⊢ (((φ ∧ χ ∧ θ) ∧ τ) → η)

Proof of Theorem syl3anl1

Step	Hyp	Ref	Expression
1		syl3anl1.1	. . 3 ⊢ (φ → ψ)
2	1	3anim1i 1138	. 2 ⊢ ((φ ∧ χ ∧ θ) → (ψ ∧ χ ∧ θ))
3		syl3anl1.2	. 2 ⊢ (((ψ ∧ χ ∧ θ) ∧ τ) → η)
4	2, 3	sylan 457	1 ⊢ (((φ ∧ χ ∧ θ) ∧ τ) → η)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by: (None)