Theorem nfrall2-P6 743

Description: ENF Over Universal Quantifier (different variable).

See nfrall2w-P6 694 for a version that requires only FOL axioms.

Hypothesis

Ref	Expression
nfrall2-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrall2-P6	⊢ Ⅎ𝑥∀𝑦𝜑

Distinct variable group:   𝑥,𝑦

Proof of Theorem nfrall2-P6

Step	Hyp	Ref	Expression
1		nfrall2-P6.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
2	1	nfrgen-P6 733	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	alloverim-P5.RC.GEN 592	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
4		allcomm-P6 739	. . . 4 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
5	4	rcp-NDBIEF0 240	. . 3 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
6	3, 5	syl-P3.24.RC 260	. 2 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
7	6	gennfr-P6 734	1 ⊢ Ⅎ𝑥∀𝑦𝜑

Syntax hints:  ∀wff-forall 8  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:  cbvall-P6-L1  750  psuball2v-P6-L1  795