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Theorem bj-nfcf 34678
Description: Version of df-nfc 2607 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.
StepHypRef Expression
1 df-nfc 2607 . 2  |-  ( F/_ x A  <->  A. z F/ x  z  e.  A )
2 bj-nfcf.nf . . . . . 6  |-  F/_ y A
32nfcri 2612 . . . . 5  |-  F/ y  z  e.  A
43nfnf 1950 . . . 4  |-  F/ y F/ x  z  e.  A
54sb8 2168 . . 3  |-  ( A. z F/ x  z  e.  A  <->  A. y [ y  /  z ] F/ x  z  e.  A
)
6 bj-sbnf 34598 . . . . 5  |-  ( [ y  /  z ] F/ x  z  e.  A  <->  F/ x [ y  /  z ] z  e.  A )
7 clelsb3 2578 . . . . . 6  |-  ( [ y  /  z ] z  e.  A  <->  y  e.  A )
87nfbii 1645 . . . . 5  |-  ( F/ x [ y  / 
z ] z  e.  A  <->  F/ x  y  e.  A )
96, 8bitri 249 . . . 4  |-  ( [ y  /  z ] F/ x  z  e.  A  <->  F/ x  y  e.  A )
109albii 1641 . . 3  |-  ( A. y [ y  /  z ] F/ x  z  e.  A  <->  A. y F/ x  y  e.  A )
115, 10bitri 249 . 2  |-  ( A. z F/ x  z  e.  A  <->  A. y F/ x  y  e.  A )
121, 11bitri 249 1  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1393   F/wnf 1617   [wsb 1740    e. wcel 1819   F/_wnfc 2605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741  df-cleq 2449  df-clel 2452  df-nfc 2607
This theorem is referenced by: (None)
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