Theorem nfnfc 2936

Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2403. (Revised by Wolf Lammen, 10-Dec-2019.)

Hypothesis

Ref	Expression
nfnfc.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfnfc	⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Proof of Theorem nfnfc

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nfc 2911	. 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴)
2		nfnfc.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3		df-nfc 2911	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
4	2, 3	mpbi 233	. . . . 5 ⊢ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴
5	4	spi 2219	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
6	5	nfnf 2358	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
7	6	nfal 2355	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
8	1, 7	nfxfr 1882	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴

This proof depends on syntax axioms:  ∀wal 1567  Ⅎwnf 1812   ∈ wcel 2142  Ⅎwnfc 2909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-11 2191  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-nf 1813  df-nfc 2911

This theorem is used by: (None)