Theorem ndore-P3.12.CL 247

Description: Closed Form of ndore-P3.12 177. †

Assertion

Ref	Expression
ndore-P3.12.CL	⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓)) → 𝜒)

Proof of Theorem ndore-P3.12.CL

Step	Hyp	Ref	Expression
1		rcp-NDASM4of4 201	. . 3 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ 𝜑) → 𝜑)
2		rcp-NDASM1of4 198	. . 3 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ 𝜑) → (𝜑 → 𝜒))
3	1, 2	ndime-P3.6 171	. 2 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ 𝜑) → 𝜒)
4		rcp-NDASM4of4 201	. . 3 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ 𝜓) → 𝜓)
5		rcp-NDASM2of4 199	. . 3 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ 𝜓) → (𝜓 → 𝜒))
6	4, 5	ndime-P3.6 171	. 2 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ 𝜓) → 𝜒)
7		rcp-NDASM3of3 197	. 2 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓))
8	3, 6, 7	rcp-NDORE4 237	1 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒) ∧ (𝜑 ∨ 𝜓)) → 𝜒)

Syntax hints:   → wff-imp 10   ∨ wff-or 144   ∧ wff-rcp-AND3 160   ∧ wff-rcp-AND4 162

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  df-rcp-AND4 163

This theorem is referenced by: (None)