Theorem oroverand-P4.27-L3 462

Description: Lemma for oroverand-P4.27b 464. †

Assertion

Ref	Expression
oroverand-P4.27-L3	⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Proof of Theorem oroverand-P4.27-L3

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	1	ndorir-P3.11 176	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ 𝜑) → (𝜑 ∨ 𝜒))
4	2, 3	ndandi-P3.7 172	. 2 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ 𝜑) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
5		rcp-NDASM2of2 194	. . . . 5 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜓 ∧ 𝜒)) → (𝜓 ∧ 𝜒))
6	5	ndander-P3.9 174	. . . 4 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜓 ∧ 𝜒)) → 𝜓)
7	6	ndoril-P3.10 175	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜓))
8	5	ndandel-P3.8 173	. . . 4 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜓 ∧ 𝜒)) → 𝜒)
9	8	ndoril-P3.10 175	. . 3 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜓 ∧ 𝜒)) → (𝜑 ∨ 𝜒))
10	7, 9	ndandi-P3.7 172	. 2 ⊢ (((𝜑 ∨ (𝜓 ∧ 𝜒)) ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
11		rcp-NDASM1of1 192	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
12	4, 10, 11	rcp-NDORE2 235	1 ⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

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

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

This theorem is referenced by:  oroverand-P4.27b  464