Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-nfs1t Structured version   Visualization version   Unicode version

Theorem wl-nfs1t 31871
Description: If  y is not free in  ph,  x is not free in  [ y  /  x ] ph. Closed form of nfs1 2194. (Contributed by Wolf Lammen, 27-Jul-2019.)
Assertion
Ref Expression
wl-nfs1t  |-  ( F/ y ph  ->  F/ x [ y  /  x ] ph )

Proof of Theorem wl-nfs1t
StepHypRef Expression
1 sbequ12r 2084 . . . . . 6  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
21equcoms 1864 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  ph ) )
32sps 1943 . . . 4  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  <->  ph ) )
43drnf1 2163 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x [
y  /  x ] ph 
<->  F/ y ph )
)
54biimprd 227 . 2  |-  ( A. x  x  =  y  ->  ( F/ y ph  ->  F/ x [ y  /  x ] ph ) )
6 nfsb2 2190 . . 3  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
76a1d 26 . 2  |-  ( -. 
A. x  x  =  y  ->  ( F/ y ph  ->  F/ x [ y  /  x ] ph ) )
85, 7pm2.61i 168 1  |-  ( F/ y ph  ->  F/ x [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188   A.wal 1442   F/wnf 1667   [wsb 1797
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-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798
This theorem is referenced by:  wl-sb8t  31880
  Copyright terms: Public domain W3C validator