Theorem oroverim-P4.28-L1 465

Description: Lemma for oroverim-P4.28a 467 and oroverimint-P4.28c 470. †

Assertion

Ref	Expression
oroverim-P4.28-L1	⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))

Proof of Theorem oroverim-P4.28-L1

Step	Hyp	Ref	Expression
1		trueie-P4.22a 444	. . . . . 6 ⊢ ((⊤ → 𝜑) ↔ 𝜑)
2	1	bisym-P3.33b.RC 299	. . . . 5 ⊢ (𝜑 ↔ (⊤ → 𝜑))
3	2	suborl-P3.43a.RC 347	. . . 4 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((⊤ → 𝜑) ∨ (𝜓 → 𝜒)))
4	3	rcp-NDBIEF0 240	. . 3 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((⊤ → 𝜑) ∨ (𝜓 → 𝜒)))
5	4	joinimor-P4.8c 403	. 2 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((⊤ ∧ 𝜓) → (𝜑 ∨ 𝜒)))
6		idandtruel-P4.19a 438	. . . 4 ⊢ ((⊤ ∧ 𝜓) ↔ 𝜓)
7	6	subiml-P3.40a.RC 326	. . 3 ⊢ (((⊤ ∧ 𝜓) → (𝜑 ∨ 𝜒)) ↔ (𝜓 → (𝜑 ∨ 𝜒)))
8	7	rcp-NDBIEF0 240	. 2 ⊢ (((⊤ ∧ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜓 → (𝜑 ∨ 𝜒)))
9		ndorir-P3.11.CL 246	. . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜒))
10	9	idandthml-P4.23a 446	. . . 4 ⊢ (((𝜑 → (𝜑 ∨ 𝜒)) ∧ (𝜓 → (𝜑 ∨ 𝜒))) ↔ (𝜓 → (𝜑 ∨ 𝜒)))
11	10	rcp-NDBIER0 241	. . 3 ⊢ ((𝜓 → (𝜑 ∨ 𝜒)) → ((𝜑 → (𝜑 ∨ 𝜒)) ∧ (𝜓 → (𝜑 ∨ 𝜒))))
12	11	joinimandinc-P4.8a 397	. 2 ⊢ ((𝜓 → (𝜑 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) → ((𝜑 ∨ 𝜒) ∨ (𝜑 ∨ 𝜒))))
13		idempotor-P4.25b 451	. . . 4 ⊢ (((𝜑 ∨ 𝜒) ∨ (𝜑 ∨ 𝜒)) ↔ (𝜑 ∨ 𝜒))
14	13	subimr-P3.40b.RC 328	. . 3 ⊢ (((𝜑 ∨ 𝜓) → ((𝜑 ∨ 𝜒) ∨ (𝜑 ∨ 𝜒))) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
15	14	rcp-NDBIEF0 240	. 2 ⊢ (((𝜑 ∨ 𝜓) → ((𝜑 ∨ 𝜒) ∨ (𝜑 ∨ 𝜒))) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
16	5, 8, 12, 15	tsyl-P4.15.RC 427	1 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144  ⊤wff-true 153

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  oroverimint-P4.28c  470  oroverbiint-P4.28d  471