Theorem 3anandirs 1284

Description: Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.)

Hypothesis

Ref	Expression
3anandirs.1	⊢ (((φ ∧ θ) ∧ (ψ ∧ θ) ∧ (χ ∧ θ)) → τ)

Assertion

Ref	Expression
3anandirs	⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ)

Proof of Theorem 3anandirs

Step	Hyp	Ref	Expression
1		simpl1 958	. 2 ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → φ)
2		simpr 447	. 2 ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → θ)
3		simpl2 959	. 2 ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → ψ)
4		simpl3 960	. 2 ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → χ)
5		3anandirs.1	. 2 ⊢ (((φ ∧ θ) ∧ (ψ ∧ θ) ∧ (χ ∧ θ)) → τ)
6	1, 2, 3, 2, 4, 2, 5	syl222anc 1198	1 ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ)

Syntax hints:   → wi 4   ∧ wa 358   ∧ 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)