Theorem nfrgen-P7 928

Description: ENF In ⇒ General For. †

Hypotheses

Ref	Expression
nfrgen-P7.1	⊢ Ⅎ𝑥𝜑
nfrgen-P7.2	⊢ (𝛾 → 𝜑)

Assertion

Ref	Expression
nfrgen-P7	⊢ (𝛾 → ∀𝑥𝜑)

Proof of Theorem nfrgen-P7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfrgen-P7.2	. . 3 ⊢ (𝛾 → 𝜑)
2		nfrgen-P7.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3	2	psubnfrv-P7 927	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	3	bisym-P3.33b.RC 299	. . 3 ⊢ (𝜑 ↔ [𝑦 / 𝑥]𝜑)
5	1, 4	subimr2-P4.RC 543	. 2 ⊢ (𝛾 → [𝑦 / 𝑥]𝜑)
6	5	ndalli-P7.17.VR12of2 866	1 ⊢ (𝛾 → ∀𝑥𝜑)

Syntax hints:  term-obj 1  ∀wff-forall 8   → wff-imp 10  Ⅎwff-nfree 681  [wff-psub 714

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  df-psub-D6.2 716

This theorem is referenced by:  nfrgen-P7.RC  929  nfrgen-P7.CL  930  nfrgen-P8  1076  nfrgen-P7.VR  1077