Theorem rcp-NDORE1 234

Description: ∨ Elimination Recipe. †

Hypotheses

Ref	Expression
rcp-NDORE1.1	⊢ (𝜑 → 𝜒)
rcp-NDORE1.2	⊢ (𝜓 → 𝜒)
rcp-NDORE1.3	⊢ (𝜑 ∨ 𝜓)

Assertion

Ref	Expression
rcp-NDORE1	⊢ 𝜒

Proof of Theorem rcp-NDORE1

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ ((⊤ ∧ 𝜑) → 𝜑)
2		rcp-NDORE1.1	. . . . 5 ⊢ (𝜑 → 𝜒)
3	2	rcp-NDIMP0addall 207	. . . 4 ⊢ ((⊤ ∧ 𝜑) → (𝜑 → 𝜒))
4	1, 3	ndime-P3.6 171	. . 3 ⊢ ((⊤ ∧ 𝜑) → 𝜒)
5		rcp-NDASM2of2 194	. . . 4 ⊢ ((⊤ ∧ 𝜓) → 𝜓)
6		rcp-NDORE1.2	. . . . 5 ⊢ (𝜓 → 𝜒)
7	6	rcp-NDIMP0addall 207	. . . 4 ⊢ ((⊤ ∧ 𝜓) → (𝜓 → 𝜒))
8	5, 7	ndime-P3.6 171	. . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜒)
9		rcp-NDORE1.3	. . . 4 ⊢ (𝜑 ∨ 𝜓)
10	9	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ∨ 𝜓))
11	4, 8, 10	ndore-P3.12 177	. 2 ⊢ (⊤ → 𝜒)
12	11	ndtruee-P3.18 183	1 ⊢ 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

This theorem is referenced by: (None)