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Theorem nfnfc 2638
Description: Hypothesis builder for  F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 1968. (Revised by Wolf Lammen, 10-Dec-2019.)
Hypothesis
Ref Expression
nfnfc.1  |-  F/_ x A
Assertion
Ref Expression
nfnfc  |-  F/ x F/_ y A

Proof of Theorem nfnfc
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-nfc 2617 . 2  |-  ( F/_ y A  <->  A. z F/ y  z  e.  A )
2 nfnfc.1 . . . . 5  |-  F/_ x A
3 nfcr 2620 . . . . 5  |-  ( F/_ x A  ->  F/ x  z  e.  A )
42, 3ax-mp 5 . . . 4  |-  F/ x  z  e.  A
54nfnf 1896 . . 3  |-  F/ x F/ y  z  e.  A
65nfal 1894 . 2  |-  F/ x A. z F/ y  z  e.  A
71, 6nfxfr 1625 1  |-  F/ x F/_ y A
Colors of variables: wff setvar class
Syntax hints:   A.wal 1377   F/wnf 1599    e. wcel 1767   F/_wnfc 2615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597  df-nf 1600  df-nfc 2617
This theorem is referenced by: (None)
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