Theorem oroverim-P4.28-L2 466

Description: Lemma for oroverim-P4.28a 467.

Assertion

Ref	Expression
oroverim-P4.28-L2	⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒)))

Proof of Theorem oroverim-P4.28-L2

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . 3 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ 𝜑) → 𝜑)
2	1	ndorir-P3.11 176	. 2 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ 𝜑) → (𝜑 ∨ (𝜓 → 𝜒)))
3		rcp-NDASM3of3 197	. . . . . . 7 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑 ∧ 𝜓) → 𝜓)
4	3	ndoril-P3.10 175	. . . . . 6 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜓))
5		rcp-NDASM1of3 195	. . . . . 6 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑 ∧ 𝜓) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
6	4, 5	ndime-P3.6 171	. . . . 5 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜒))
7		rcp-NDASM2of3 196	. . . . 5 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑 ∧ 𝜓) → ¬ 𝜑)
8	6, 7	profeliml-P4.5a 385	. . . 4 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑 ∧ 𝜓) → 𝜒)
9	8	rcp-NDIMI3 225	. . 3 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑) → (𝜓 → 𝜒))
10	9	ndoril-P3.10 175	. 2 ⊢ ((((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ∧ ¬ 𝜑) → (𝜑 ∨ (𝜓 → 𝜒)))
11		ndexclmid-P3.16.AC 251	. 2 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ ¬ 𝜑))
12	2, 10, 11	rcp-NDORE2 235	1 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒)))

Syntax hints:  ¬ wff-neg 9   → 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:  oroverim-P4.28a  467