Theorem andassoc-P3.36-L1 315

Description: Lemma for andassoc-P3.36 317. †

Assertion

Ref	Expression
andassoc-P3.36-L1	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → (𝜑 ∧ (𝜓 ∧ 𝜒)))

Proof of Theorem andassoc-P3.36-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM1of1 192	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∧ 𝜒))
2	1	ndander-P3.9 174	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → (𝜑 ∧ 𝜓))
3	2	ndander-P3.9 174	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜑)
4	2	ndandel-P3.8 173	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜓)
5	1	ndandel-P3.8 173	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜒)
6	4, 5	ndandi-P3.7 172	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → (𝜓 ∧ 𝜒))
7	3, 6	ndandi-P3.7 172	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → (𝜑 ∧ (𝜓 ∧ 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  andassoc-P3.36  317