Theorem nfnf 2324

Description: If 𝑥 is not free in 𝜑, then it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypothesis

Ref	Expression
nfnf.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfnf	⊢ Ⅎ𝑥Ⅎ𝑦𝜑

Proof of Theorem nfnf

Step	Hyp	Ref	Expression
1		df-nf 1788	. 2 ⊢ (Ⅎ𝑦𝜑 ↔ (∃𝑦𝜑 → ∀𝑦𝜑))
2		nfnf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	nfex 2322	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜑
4	2	nfal 2321	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜑
5	3, 4	nfim 1900	. 2 ⊢ Ⅎ𝑥(∃𝑦𝜑 → ∀𝑦𝜑)
6	1, 5	nfxfr 1856	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝜑

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by:  nfnfc  2918  bj-nfcf  35038