Theorem orassoc-P3.38-L2 321

Description: Lemma for orassoc-P3.38 322. †

Assertion

Ref	Expression
orassoc-P3.38-L2	⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) ∨ 𝜒))

Proof of Theorem orassoc-P3.38-L2

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ 𝜑) → 𝜑)
2	1	ndorir-P3.11 176	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ 𝜑) → (𝜑 ∨ 𝜓))
3	2	ndorir-P3.11 176	. 2 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ 𝜑) → ((𝜑 ∨ 𝜓) ∨ 𝜒))
4		rcp-NDASM3of3 197	. . . . 5 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒) ∧ 𝜓) → 𝜓)
5	4	ndoril-P3.10 175	. . . 4 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒) ∧ 𝜓) → (𝜑 ∨ 𝜓))
6	5	ndorir-P3.11 176	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒) ∧ 𝜓) → ((𝜑 ∨ 𝜓) ∨ 𝜒))
7		rcp-NDASM3of3 197	. . . 4 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒) ∧ 𝜒) → 𝜒)
8	7	ndoril-P3.10 175	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒) ∧ 𝜒) → ((𝜑 ∨ 𝜓) ∨ 𝜒))
9		rcp-NDASM2of2 194	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒)) → (𝜓 ∨ 𝜒))
10	6, 8, 9	rcp-NDORE3 236	. 2 ⊢ (((𝜑 ∨ (𝜓 ∨ 𝜒)) ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) ∨ 𝜒))
11		rcp-NDASM1of1 192	. 2 ⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∨ 𝜒)))
12	3, 10, 11	rcp-NDORE2 235	1 ⊢ ((𝜑 ∨ (𝜓 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) ∨ 𝜒))

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144   ∧ wff-rcp-AND3 160

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-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  orassoc-P3.38  322