Theorem nfnfc 2913

Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2369. (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 2883	. 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴)
2		nfnfc.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3		df-nfc 2883	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
4	2, 3	mpbi 229	. . . . 5 ⊢ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴
5	4	spi 2175	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
6	5	nfnf 2317	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
7	6	nfal 2314	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
8	1, 7	nfxfr 1853	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Syntax hints:  ∀wal 1537  Ⅎwnf 1783   ∈ wcel 2104  Ⅎwnfc 2881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-11 2152  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784  df-nfc 2883

This theorem is referenced by: (None)