Theorem nfal 1629

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 1563. (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 1514	. . . . . 6 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2	1	biimpi 120	. . . . 5 ⊢ (Ⅎ𝑥𝜑 → ∀𝑥(𝜑 → ∀𝑥𝜑))
3	2	alimi 1508	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
4		ax-7 1501	. . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
5		ax-5 1500	. . . . . 6 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
6		ax-7 1501	. . . . . 6 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
7	5, 6	syl6 33	. . . . 5 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
8	7	alimi 1508	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
9	3, 4, 8	3syl 17	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
10		df-nf 1514	. . 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 1504	1 ⊢ Ⅎ𝑥∀𝑦𝜑

Syntax hints:   → wi 4  ∀wal 1400  Ⅎwnf 1513

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 1500  ax-7 1501  ax-gen 1502

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  nfnf  1630  nfa2  1632  aaan  1640  cbv3  1795  cbv2  1802  nfald  1813  cbval2  1977  nfsb4t  2074  nfeuv  2104  mo23  2128  bm1.1  2223  nfnfc1  2395  nfnfc  2399  nfeq  2400  nfabdw  2411  sbcnestgf  3199  dfnfc2  3948  nfdisjv  4113  nfdisj1  4114  nffr  4489  uchoice  6361  modom  7098  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  exmidunben  13295  bdsepnft  16827  bdsepnfALT  16829  setindft  16905  strcollnft  16924  pw1nct  16947