Theorem nfnfc 2760

Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2234. (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 2740	. 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴)
2		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcr 2743	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴)
4	2, 3	ax-mp 5	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
5	4	nfnf 2144	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
6	5	nfal 2139	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
7	1, 6	nfxfr 1771	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Syntax hints:  ∀wal 1473  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-tru 1478  df-ex 1696  df-nf 1701  df-nfc 2740

This theorem is referenced by: (None)