Theorem ndnfrneg-P7.2 827

Description: Natural Deduction: Effective Non-Freeness Rule 2.

If '𝑥' is (conditionally) effectively not free in '𝜑', then '𝑥' is (conditionally) effectively not free in '¬ 𝜑'.

Hypothesis

Ref	Expression
ndnfrneg-P7.2.1	⊢ (𝛾 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
ndnfrneg-P7.2	⊢ (𝛾 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem ndnfrneg-P7.2

Step	Hyp	Ref	Expression
1		ndnfrneg-P7.2.1	. 2 ⊢ (𝛾 → Ⅎ𝑥𝜑)
2		nfrneg-P6 688	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
3	2	rcp-NDBIER0 241	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
4	3	imsubr-P3.28b.RC 270	. 2 ⊢ ((𝛾 → Ⅎ𝑥𝜑) → (𝛾 → Ⅎ𝑥 ¬ 𝜑))
5	1, 4	rcp-NDIME0 228	1 ⊢ (𝛾 → Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16

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-nfree-D6.1 682

This theorem is referenced by:  ndnfrneg-P7.2.RC  875  ndnfrneg-P7.2.CL  904