Theorem sepimorl-P4.9b 409

Description: Separate Left Disjunction from Implication. †

Hypothesis

Ref	Expression
sepimorl-P4.9b.1	⊢ (𝛾 → ((𝜑 ∨ 𝜓) → 𝜒))

Assertion

Ref	Expression
sepimorl-P4.9b	⊢ (𝛾 → ((𝜑 → 𝜒) ∧ (𝜓 → 𝜒)))

Proof of Theorem sepimorl-P4.9b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . . 5 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2	1	ndorir-P3.11 176	. . . 4 ⊢ ((𝛾 ∧ 𝜑) → (𝜑 ∨ 𝜓))
3		sepimorl-P4.9b.1	. . . . 5 ⊢ (𝛾 → ((𝜑 ∨ 𝜓) → 𝜒))
4	3	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ 𝜑) → ((𝜑 ∨ 𝜓) → 𝜒))
5	2, 4	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜒)
6	5	rcp-NDIMI2 224	. 2 ⊢ (𝛾 → (𝜑 → 𝜒))
7		rcp-NDASM2of2 194	. . . . 5 ⊢ ((𝛾 ∧ 𝜓) → 𝜓)
8	7	ndoril-P3.10 175	. . . 4 ⊢ ((𝛾 ∧ 𝜓) → (𝜑 ∨ 𝜓))
9	3	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ 𝜓) → ((𝜑 ∨ 𝜓) → 𝜒))
10	8, 9	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ 𝜓) → 𝜒)
11	10	rcp-NDIMI2 224	. 2 ⊢ (𝛾 → (𝜓 → 𝜒))
12	6, 11	ndandi-P3.7 172	1 ⊢ (𝛾 → ((𝜑 → 𝜒) ∧ (𝜓 → 𝜒)))

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

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

This theorem is referenced by:  sepimorl-P4.9b.RC  410  sepimorl-P4.9b.CL  411