Theorem suborl-P3.43a-L1 345

Description: Lemma for suborl-P3.43a 346. †

Hypothesis

Ref	Expression
suborl-P3.43a-L1.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
suborl-P3.43a-L1	⊢ (𝛾 → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜒)))

Proof of Theorem suborl-P3.43a-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → 𝜑)
2		suborl-P3.43a-L1.1	. . . . . . 7 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3	2	rcp-NDIMP1add2 212	. . . . . 6 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → (𝜑 ↔ 𝜓))
4	3	ndbief-P3.14 179	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → (𝜑 → 𝜓))
5	1, 4	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → 𝜓)
6	5	ndorir-P3.11 176	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → (𝜓 ∨ 𝜒))
7		rcp-NDASM3of3 197	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜒) → 𝜒)
8	7	ndoril-P3.10 175	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜒) → (𝜓 ∨ 𝜒))
9		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ 𝜒))
10	6, 8, 9	rcp-NDORE3 236	. 2 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒))
11	10	rcp-NDIMI2 224	1 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜒)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∨ 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-rcp-AND3 161

This theorem is referenced by:  suborl-P3.43a  346