Theorem bj-nfcf 34366

Description: Version of df-nfc 2938 with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 2-May-2019.)

Hypothesis

Ref	Expression
bj-nfcf.nf	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
bj-nfcf	⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem bj-nfcf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nfc 2938	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
2		bj-nfcf.nf	. . . . . 6 ⊢ Ⅎ𝑦𝐴
3	2	nfcri 2943	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
4	3	nfnf 2334	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑥 𝑧 ∈ 𝐴
5	4	sb8v 2362	. . 3 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴)
6		bj-sbnf 34279	. . . . 5 ⊢ ([𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴)
7		clelsb3 2917	. . . . . 6 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
8	7	nfbii 1853	. . . . 5 ⊢ (Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴)
9	6, 8	bitri 278	. . . 4 ⊢ ([𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴)
10	9	albii 1821	. . 3 ⊢ (∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
11	5, 10	bitri 278	. 2 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
12	1, 11	bitri 278	1 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  Ⅎwnfc 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-10 2142  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2870  df-nfc 2938

This theorem is referenced by: (None)