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

Theorem hbs1 2057
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 1998 . 2  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
2 hbsb2 2010 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
31, 2pm2.61i 156 1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   [wsb 1638
This theorem is referenced by:  nfs1v  2058  hbab1  2285  sb5ALT  28587  2sb5ndVD  29002  sb5ALTVD  29005  2sb5ndALT  29025
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  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
  Copyright terms: Public domain W3C validator