Theorem syld3an3 1227

Description: A syllogism inference. (Contributed by NM, 20-May-2007.)

Hypotheses

Ref	Expression
syld3an3.1	⊢ ((φ ∧ ψ ∧ χ) → θ)
syld3an3.2	⊢ ((φ ∧ ψ ∧ θ) → τ)

Assertion

Ref	Expression
syld3an3	⊢ ((φ ∧ ψ ∧ χ) → τ)

Proof of Theorem syld3an3

Step	Hyp	Ref	Expression
1		simp1 955	. 2 ⊢ ((φ ∧ ψ ∧ χ) → φ)
2		simp2 956	. 2 ⊢ ((φ ∧ ψ ∧ χ) → ψ)
3		syld3an3.1	. 2 ⊢ ((φ ∧ ψ ∧ χ) → θ)
4		syld3an3.2	. 2 ⊢ ((φ ∧ ψ ∧ θ) → τ)
5	1, 2, 3, 4	syl3anc 1182	1 ⊢ ((φ ∧ ψ ∧ χ) → τ)

Syntax hints:   → wi 4   ∧ 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:  syld3an1  1228  syld3an2  1229  resin  5308