Theorem sepimorr-P4.9c 412

Description: Separate Right Disjunction from Implication.

Hypothesis

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

Assertion

Ref	Expression
sepimorr-P4.9c	⊢ (𝛾 → ((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)))

Proof of Theorem sepimorr-P4.9c

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ ((𝛾 ∧ 𝜓) → 𝜓)
2	1	axL1-P3.21 252	. . 3 ⊢ ((𝛾 ∧ 𝜓) → (𝜑 → 𝜓))
3	2	ndorir-P3.11 176	. 2 ⊢ ((𝛾 ∧ 𝜓) → ((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)))
4		rcp-NDASM3of3 197	. . . . . 6 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ 𝜑) → 𝜑)
5		sepimorr-P4.9c.1	. . . . . . 7 ⊢ (𝛾 → (𝜑 → (𝜓 ∨ 𝜒)))
6	5	rcp-NDIMP1add2 212	. . . . . 6 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ 𝜑) → (𝜑 → (𝜓 ∨ 𝜒)))
7	4, 6	ndime-P3.6 171	. . . . 5 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ 𝜑) → (𝜓 ∨ 𝜒))
8		rcp-NDASM2of3 196	. . . . 5 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ 𝜑) → ¬ 𝜓)
9	7, 8	profeliml-P4.5a 385	. . . 4 ⊢ ((𝛾 ∧ ¬ 𝜓 ∧ 𝜑) → 𝜒)
10	9	rcp-NDIMI3 225	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜓) → (𝜑 → 𝜒))
11	10	ndoril-P3.10 175	. 2 ⊢ ((𝛾 ∧ ¬ 𝜓) → ((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)))
12		ndexclmid-P3.16.AC 251	. 2 ⊢ (𝛾 → (𝜓 ∨ ¬ 𝜓))
13	3, 11, 12	rcp-NDORE2 235	1 ⊢ (𝛾 → ((𝜑 → 𝜓) ∨ (𝜑 → 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:  sepimorr-P4.9c.RC  413  sepimorr-P4.9c.CL  414