Theorem bj-nfnfc 33159

Description: Remove dependency on ax-ext 2740 (and df-cleq 2753) from nfnfc 2912. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-nfnfc.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
bj-nfnfc	⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Proof of Theorem bj-nfnfc

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nfc 2891	. 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴)
2		bj-nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	bj-nfcri 33158	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4	3	nfnf 2305	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
5	4	nfal 2300	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
6	1, 5	nfxfr 1928	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Syntax hints:  ∀wal 1630  Ⅎwnf 1857   ∈ wcel 2139  Ⅎwnfc 2889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clel 2756  df-nfc 2891

This theorem is referenced by: (None)