MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfnf Unicode version

Theorem nfnf 1780
Description: If  x is not free in  ph, it is not free in  F/ y ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nf.1  |-  F/ x ph
Assertion
Ref Expression
nfnf  |-  F/ x F/ y ph

Proof of Theorem nfnf
StepHypRef Expression
1 df-nf 1535 . 2  |-  ( F/ y ph  <->  A. y
( ph  ->  A. y ph ) )
2 nf.1 . . . . . 6  |-  F/ x ph
32a1i 10 . . . . 5  |-  (  T. 
->  F/ x ph )
42nfal 1778 . . . . . 6  |-  F/ x A. y ph
54a1i 10 . . . . 5  |-  (  T. 
->  F/ x A. y ph )
63, 5nfimd 1773 . . . 4  |-  (  T. 
->  F/ x ( ph  ->  A. y ph )
)
76trud 1314 . . 3  |-  F/ x
( ph  ->  A. y ph )
87nfal 1778 . 2  |-  F/ x A. y ( ph  ->  A. y ph )
91, 8nfxfr 1560 1  |-  F/ x F/ y ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    T. wtru 1307   A.wal 1530   F/wnf 1534
This theorem is referenced by:  nfnfc  2438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-nf 1535
  Copyright terms: Public domain W3C validator