Theorem bj-nfcf 32920

Description: Version of df-nfc 2753 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.)

Hypothesis

Ref	Expression
bj-nfcf.nf	|- F/_ y A

Assertion

Ref	Expression
bj-nfcf	|- ( F/_ x A <-> A. y F/ x y e. 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 2753	. 2 |- ( F/_ x A <-> A. z F/ x z e. A )
2		bj-nfcf.nf	. . . . . 6 |- F/_ y A
3	2	nfcri 2758	. . . . 5 |- F/ y z e. A
4	3	nfnf 2158	. . . 4 |- F/ y F/ x z e. A
5	4	sb8 2424	. . 3 |- ( A. z F/ x z e. A <-> A. y [ y / z ] F/ x z e. A )
6		bj-sbnf 32828	. . . . 5 |- ( [ y / z ] F/ x z e. A <-> F/ x [ y / z ] z e. A )
7		clelsb3 2729	. . . . . 6 |- ( [ y / z ] z e. A <-> y e. A )
8	7	nfbii 1778	. . . . 5 |- ( F/ x [ y / z ] z e. A <-> F/ x y e. A )
9	6, 8	bitri 264	. . . 4 |- ( [ y / z ] F/ x z e. A <-> F/ x y e. A )
10	9	albii 1747	. . 3 |- ( A. y [ y / z ] F/ x z e. A <-> A. y F/ x y e. A )
11	5, 10	bitri 264	. 2 |- ( A. z F/ x z e. A <-> A. y F/ x y e. A )
12	1, 11	bitri 264	1 |- ( F/_ x A <-> A. y F/ x y e. A )

Syntax hints:   <-> wb 196   A.wal 1481   F/wnf 1708   [wsb 1880   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by: (None)