Theorem nfal 1587

Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1521. (Revised by GG, 25-Aug-2024.)

Hypothesis

Ref	Expression
nfal.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfal	⊢ Ⅎ𝑥∀𝑦𝜑

Proof of Theorem nfal

Step	Hyp	Ref	Expression
1		df-nf 1472	. . . . . 6 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2	1	biimpi 120	. . . . 5 ⊢ (Ⅎ𝑥𝜑 → ∀𝑥(𝜑 → ∀𝑥𝜑))
3	2	alimi 1466	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
4		ax-7 1459	. . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
5		ax-5 1458	. . . . . 6 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
6		ax-7 1459	. . . . . 6 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
7	5, 6	syl6 33	. . . . 5 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
8	7	alimi 1466	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
9	3, 4, 8	3syl 17	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
10		df-nf 1472	. . 3 ⊢ (Ⅎ𝑥∀𝑦𝜑 ↔ ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
11	9, 10	sylibr 134	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥∀𝑦𝜑)
12		nfal.1	. 2 ⊢ Ⅎ𝑥𝜑
13	11, 12	mpg 1462	1 ⊢ Ⅎ𝑥∀𝑦𝜑

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  nfnf  1588  nfa2  1590  aaan  1598  cbv3  1753  cbv2  1760  nfald  1771  cbval2  1933  nfsb4t  2030  nfeuv  2060  mo23  2083  bm1.1  2178  nfnfc1  2339  nfnfc  2343  nfeq  2344  nfabdw  2355  sbcnestgf  3132  dfnfc2  3853  nfdisjv  4018  nfdisj1  4019  nffr  4380  uchoice  6190  exmidfodomrlemr  7262  exmidfodomrlemrALT  7263  exmidunben  12583  bdsepnft  15379  bdsepnfALT  15381  setindft  15457  strcollnft  15476  pw1nct  15493