MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbsb2 Unicode version

Theorem hbsb2 1997
Description: Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hbsb2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )

Proof of Theorem hbsb2
StepHypRef Expression
1 sb4 1993 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 1963 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
32a5i 1758 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x [ y  /  x ] ph )
41, 3syl6 29 1  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   [wsb 1629
This theorem is referenced by:  nfsb2  1998  sbequi  1999  sb9i  2034  hbs1  2044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
  Copyright terms: Public domain W3C validator