Theorem nfrgencl-P7 965

Description: ENF In ⇒ General For (closed form). †

Assertion

Ref	Expression
nfrgencl-P7	⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nfrgencl-P7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ndnfrnfr-P7.12 837	. . . . . 6 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
2		rcp-NDASM1of1 192	. . . . . 6 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3	1, 2	lemma-L7.03 962	. . . . 5 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
4	3	ndbier-P3.15 180	. . . 4 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑))
5	4	import-P3.34a.RC 306	. . 3 ⊢ ((Ⅎ𝑥𝜑 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)
6	5	ndalli-P7.17.VR12of2 866	. 2 ⊢ ((Ⅎ𝑥𝜑 ∧ 𝜑) → ∀𝑥𝜑)
7	6	rcp-NDIMI2 224	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Syntax hints:  term-obj 1  ∀wff-forall 8   → wff-imp 10   ∧ wff-and 132  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:  dfnfreealtonlyif-P7  966