Theorem syl3anb 1225

Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)

Hypotheses

Ref	Expression
syl3anb.1	⊢ (φ ↔ ψ)
syl3anb.2	⊢ (χ ↔ θ)
syl3anb.3	⊢ (τ ↔ η)
syl3anb.4	⊢ ((ψ ∧ θ ∧ η) → ζ)

Assertion

Ref	Expression
syl3anb	⊢ ((φ ∧ χ ∧ τ) → ζ)

Proof of Theorem syl3anb

Step	Hyp	Ref	Expression
1		syl3anb.1	. . 3 ⊢ (φ ↔ ψ)
2		syl3anb.2	. . 3 ⊢ (χ ↔ θ)
3		syl3anb.3	. . 3 ⊢ (τ ↔ η)
4	1, 2, 3	3anbi123i 1140	. 2 ⊢ ((φ ∧ χ ∧ τ) ↔ (ψ ∧ θ ∧ η))
5		syl3anb.4	. 2 ⊢ ((ψ ∧ θ ∧ η) → ζ)
6	4, 5	sylbi 187	1 ⊢ ((φ ∧ χ ∧ τ) → ζ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ 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:  syl3anbr  1226