Theorem oroverand-P4.27-L4 463

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

Assertion

Ref	Expression
oroverand-P4.27-L4	⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒)))

Proof of Theorem oroverand-P4.27-L4

Step	Hyp	Ref	Expression
1		rcp-NDASM3of3 197	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → 𝜑)
2	1	ndorir-P3.11 176	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
3		rcp-NDASM4of4 201	. . . 4 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓 ∧ 𝜑) → 𝜑)
4	3	ndorir-P3.11 176	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓 ∧ 𝜑) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
5		rcp-NDASM3of4 200	. . . . 5 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓 ∧ 𝜒) → 𝜓)
6		rcp-NDASM4of4 201	. . . . 5 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓 ∧ 𝜒) → 𝜒)
7	5, 6	ndandi-P3.7 172	. . . 4 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))
8	7	ndoril-P3.10 175	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓 ∧ 𝜒) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
9		rcp-NDASM2of3 196	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓) → (𝜑 ∨ 𝜒))
10	4, 8, 9	rcp-NDORE4 237	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ 𝜓) → (𝜑 ∨ (𝜓 ∧ 𝜒)))
11		rcp-NDASM1of2 193	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ 𝜓))
12	2, 10, 11	rcp-NDORE3 236	1 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 ∧ 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144   ∧ wff-rcp-AND3 160   ∧ wff-rcp-AND4 162

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  df-rcp-AND4 163

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