Theorem andoveror-P4.27-L1 459

Description: Lemma for andoveror-P4.27a 461. †

Assertion

Ref	Expression
andoveror-P4.27-L1	⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))

Proof of Theorem andoveror-P4.27-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM1of3 195	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜓) → 𝜑)
2		rcp-NDASM3of3 197	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜓) → 𝜓)
3	1, 2	ndandi-P3.7 172	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜓) → (𝜑 ∧ 𝜓))
4	3	ndorir-P3.11 176	. 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
5		rcp-NDASM1of3 195	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜒) → 𝜑)
6		rcp-NDASM3of3 197	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜒) → 𝜒)
7	5, 6	ndandi-P3.7 172	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜒) → (𝜑 ∧ 𝜒))
8	7	ndoril-P3.10 175	. 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒) ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
9		rcp-NDASM2of2 194	. 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → (𝜓 ∨ 𝜒))
10	4, 8, 9	rcp-NDORE3 236	1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 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:  andoveror-P4.27a  461