MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbs1 Unicode version

Theorem hbs1 2180
Description:  x is not free in  [
y  /  x ] ph when  x and  y are distinct. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hbs1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem hbs1
StepHypRef Expression
1 ax16 2129 . 2  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
2 hbsb2 2091 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
31, 2pm2.61i 158 1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   [wsb 1658
This theorem is referenced by:  nfs1v  2181  hbab1  2419  sb5ALT  28362  2sb5ndVD  28774  sb5ALTVD  28777  2sb5ndALT  28797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
  Copyright terms: Public domain W3C validator