Theorem nfal 1598

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 1532. (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 1483	. . . . . 6 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2	1	biimpi 120	. . . . 5 ⊢ (Ⅎ𝑥𝜑 → ∀𝑥(𝜑 → ∀𝑥𝜑))
3	2	alimi 1477	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
4		ax-7 1470	. . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
5		ax-5 1469	. . . . . 6 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
6		ax-7 1470	. . . . . 6 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
7	5, 6	syl6 33	. . . . 5 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
8	7	alimi 1477	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
9	3, 4, 8	3syl 17	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
10		df-nf 1483	. . 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 1473	1 ⊢ Ⅎ𝑥∀𝑦𝜑

Syntax hints:   → wi 4  ∀wal 1370  Ⅎwnf 1482

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 1469  ax-7 1470  ax-gen 1471

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  nfnf  1599  nfa2  1601  aaan  1609  cbv3  1764  cbv2  1771  nfald  1782  cbval2  1944  nfsb4t  2041  nfeuv  2071  mo23  2094  bm1.1  2189  nfnfc1  2350  nfnfc  2354  nfeq  2355  nfabdw  2366  sbcnestgf  3144  dfnfc2  3867  nfdisjv  4032  nfdisj1  4033  nffr  4395  uchoice  6222  exmidfodomrlemr  7309  exmidfodomrlemrALT  7310  exmidunben  12739  bdsepnft  15756  bdsepnfALT  15758  setindft  15834  strcollnft  15853  pw1nct  15873