Theorem syl3an1 1215

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)

Hypotheses

Ref	Expression
syl3an1.1	⊢ (φ → ψ)
syl3an1.2	⊢ ((ψ ∧ χ ∧ θ) → τ)

Assertion

Ref	Expression
syl3an1	⊢ ((φ ∧ χ ∧ θ) → τ)

Proof of Theorem syl3an1

Step	Hyp	Ref	Expression
1		syl3an1.1	. . 3 ⊢ (φ → ψ)
2	1	3anim1i 1138	. 2 ⊢ ((φ ∧ χ ∧ θ) → (ψ ∧ χ ∧ θ))
3		syl3an1.2	. 2 ⊢ ((ψ ∧ χ ∧ θ) → τ)
4	2, 3	syl 15	1 ⊢ ((φ ∧ χ ∧ θ) → τ)

This proof depends on syntax axioms:   → wi 4   ∧ 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:  syl3an1b  1218  syl3an1br  1221  ltfintri  4467  lecadd2  6267