Theorem ndnfrall2-P7.9 834

Description: Natural Deduction: Effective Non-Freeness Rule 9.

If '𝑥' is (conditionally) effectively not free in '𝜑', then '𝑥' is also (conditionally) effectively not free in '∀𝑦𝜑', where '𝑦' is different from '𝑥' and neither '𝑥' nor '𝑦' are effectively free in the context, '𝛾'.

Hypotheses

Ref	Expression
ndnfrall2-P7.9.1	⊢ Ⅎ𝑥𝛾
ndnfrall2-P7.9.2	⊢ Ⅎ𝑦𝛾
ndnfrall2-P7.9.3	⊢ (𝛾 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
ndnfrall2-P7.9	⊢ (𝛾 → Ⅎ𝑥∀𝑦𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem ndnfrall2-P7.9

Step	Hyp	Ref	Expression
1		ndnfrall2-P7.9.1	. 2 ⊢ Ⅎ𝑥𝛾
2		ndnfrall2-P7.9.2	. 2 ⊢ Ⅎ𝑦𝛾
3		ndnfrall2-P7.9.3	. 2 ⊢ (𝛾 → Ⅎ𝑥𝜑)
4	1, 2, 3	nfrall2d-P6 819	1 ⊢ (𝛾 → Ⅎ𝑥∀𝑦𝜑)

Syntax hints:  ∀wff-forall 8   → 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  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

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-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  ndnfrall2-P7.9.VR12of2  860