Theorem 3anandirs 1470

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 1189	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜑)
2		simpr 484	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜃)
3		simpl2 1190	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓)
4		simpl3 1191	. 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 1384	1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

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 206  df-an 396  df-3an 1087

This theorem is referenced by:  leoptr  30400