Theorem syl3an1b 1218

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

Hypotheses

Ref	Expression
syl3an1b.1	⊢ (φ ↔ ψ)
syl3an1b.2	⊢ ((ψ ∧ χ ∧ θ) → τ)

Assertion

Ref	Expression
syl3an1b	⊢ ((φ ∧ χ ∧ θ) → τ)

Proof of Theorem syl3an1b

Step	Hyp	Ref	Expression
1		syl3an1b.1	. . 3 ⊢ (φ ↔ ψ)
2	1	biimpi 186	. 2 ⊢ (φ → ψ)
3		syl3an1b.2	. 2 ⊢ ((ψ ∧ χ ∧ θ) → τ)
4	2, 3	syl3an1 1215	1 ⊢ ((φ ∧ χ ∧ θ) → τ)

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: (None)