Theorem nfnfc 2938

Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2404. (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 2913	. 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴)
2		nfnfc.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3		df-nfc 2913	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
4	2, 3	mpbi 233	. . . . 5 ⊢ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴
5	4	spi 2224	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
6	5	nfnf 2360	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
7	6	nfal 2357	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
8	1, 7	nfxfr 1886	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴

This proof depends on syntax axioms:  ∀wal 1568  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-nfc 2913

This theorem is used by: (None)