Theorem syl3an 1224

Description: A triple syllogism inference. (Contributed by NM, 13-May-2004.)

Hypotheses

Ref	Expression
syl3an.1	⊢ (φ → ψ)
syl3an.2	⊢ (χ → θ)
syl3an.3	⊢ (τ → η)
syl3an.4	⊢ ((ψ ∧ θ ∧ η) → ζ)

Assertion

Ref	Expression
syl3an	⊢ ((φ ∧ χ ∧ τ) → ζ)

Proof of Theorem syl3an

Step	Hyp	Ref	Expression
1		syl3an.1	. . 3 ⊢ (φ → ψ)
2		syl3an.2	. . 3 ⊢ (χ → θ)
3		syl3an.3	. . 3 ⊢ (τ → η)
4	1, 2, 3	3anim123i 1137	. 2 ⊢ ((φ ∧ χ ∧ τ) → (ψ ∧ θ ∧ η))
5		syl3an.4	. 2 ⊢ ((ψ ∧ θ ∧ η) → ζ)
6	4, 5	syl 15	1 ⊢ ((φ ∧ χ ∧ τ) → ζ)

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:  peano5  4410  spfininduct  4541  eloprabga  5579  clos1induct  5881