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Theorem nfnfc 2601
Description: Hypothesis builder for  F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2091. (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 2581 . 2  |-  ( F/_ y A  <->  A. z F/ y  z  e.  A )
2 nfnfc.1 . . . . 5  |-  F/_ x A
3 nfcr 2584 . . . . 5  |-  ( F/_ x A  ->  F/ x  z  e.  A )
42, 3ax-mp 5 . . . 4  |-  F/ x  z  e.  A
54nfnf 2032 . . 3  |-  F/ x F/ y  z  e.  A
65nfal 2030 . 2  |-  F/ x A. z F/ y  z  e.  A
71, 6nfxfr 1696 1  |-  F/ x F/_ y A
Colors of variables: wff setvar class
Syntax hints:   A.wal 1442   F/wnf 1667    e. wcel 1887   F/_wnfc 2579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-nfc 2581
This theorem is referenced by: (None)
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