Theorem nfnf1 1597

Description: 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1514	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nfa1 1594	. 2 ⊢ Ⅎ𝑥∀𝑥(𝜑 → ∀𝑥𝜑)
3	1, 2	nfxfr 1527	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1400  Ⅎwnf 1513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ial 1587

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

This theorem is used by:  nfimd  1638  nfnt  1708  nfald  1813  equs5or  1883  sbcomxyyz  2032  nfsb4t  2074  nfnfc1  2395  nfabdw  2411  sbcnestgf  3199  dfnfc2  3953  bdsepnft  16913  setindft  16991  strcollnft  17010