Theorem rcp-NDNEGI1 218

Description: ¬ Introduction Recipe. †

Hypotheses

Ref	Expression
rcp-NDNEGI1.1	⊢ (𝛾₁ → 𝜑)
rcp-NDNEGI1.2	⊢ (𝛾₁ → ¬ 𝜑)

Assertion

Ref	Expression
rcp-NDNEGI1	⊢ ¬ 𝛾₁

Proof of Theorem rcp-NDNEGI1

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ ((⊤ ∧ 𝛾₁) → 𝛾₁)
2		rcp-NDNEGI1.1	. . . . 5 ⊢ (𝛾₁ → 𝜑)
3	2	rcp-NDIMP0addall 207	. . . 4 ⊢ ((⊤ ∧ 𝛾₁) → (𝛾₁ → 𝜑))
4	1, 3	ndime-P3.6 171	. . 3 ⊢ ((⊤ ∧ 𝛾₁) → 𝜑)
5		rcp-NDNEGI1.2	. . . . 5 ⊢ (𝛾₁ → ¬ 𝜑)
6	5	rcp-NDIMP0addall 207	. . . 4 ⊢ ((⊤ ∧ 𝛾₁) → (𝛾₁ → ¬ 𝜑))
7	1, 6	ndime-P3.6 171	. . 3 ⊢ ((⊤ ∧ 𝛾₁) → ¬ 𝜑)
8	4, 7	ndnegi-P3.3 168	. 2 ⊢ (⊤ → ¬ 𝛾₁)
9	8	ndtruee-P3.18 183	1 ⊢ ¬ 𝛾₁

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132  ⊤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-true-D2.4 155

This theorem is referenced by:  ncontra-P4.1  366  rcp-RAA1  515  example-E7.1b  1075