Theorem bj-nfcf 36107

Description: Version of df-nfc 2884 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 2884	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
2		bj-nfcf.nf	. . . . . 6 ⊢ Ⅎ𝑦𝐴
3	2	nfcri 2889	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
4	3	nfnf 2318	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑥 𝑧 ∈ 𝐴
5	4	sb8f 2348	. . 3 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴)
6		bj-sbnf 36023	. . . . 5 ⊢ ([𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴)
7		clelsb1 2859	. . . . . 6 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
8	7	nfbii 1853	. . . . 5 ⊢ (Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴)
9	6, 8	bitri 275	. . . 4 ⊢ ([𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴)
10	9	albii 1820	. . 3 ⊢ (∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
11	5, 10	bitri 275	. 2 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
12	1, 11	bitri 275	1 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 205  ∀wal 1538  Ⅎwnf 1784  [wsb 2066   ∈ wcel 2105  Ⅎwnfc 2882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-10 2136  ax-11 2153  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1781  df-nf 1785  df-sb 2067  df-clel 2809  df-nfc 2884

This theorem is referenced by: (None)