Theorem nfnfcALT 2761

Description: Alternate proof of nfnfc 2760. Shorter but requiring more axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfnfc.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfnfcALT	⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Proof of Theorem nfnfcALT

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	2	nfcri 2745	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4	3	nfnf 2144	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
5	4	nfal 2139	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
6	1, 5	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  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740

This theorem is referenced by: (None)