Theorem 3anandis 1283

Description: Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)

Hypothesis

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

Assertion

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

Proof of Theorem 3anandis

Step	Hyp	Ref	Expression
1		simpl 443	. 2 ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → φ)
2		simpr1 961	. 2 ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → ψ)
3		simpr2 962	. 2 ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → χ)
4		simpr3 963	. 2 ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → θ)
5		3anandis.1	. 2 ⊢ (((φ ∧ ψ) ∧ (φ ∧ χ) ∧ (φ ∧ θ)) → τ)
6	1, 2, 1, 3, 1, 4, 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)