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

Theorem hbsb 2290
Description: If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by NM, 12-Aug-1993.)
Hypothesis
Ref Expression
hbsb.1  |-  ( ph  ->  A. z ph )
Assertion
Ref Expression
hbsb  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem hbsb
StepHypRef Expression
1 hbsb.1 . . . 4  |-  ( ph  ->  A. z ph )
21nfi 1682 . . 3  |-  F/ z
ph
32nfsb 2289 . 2  |-  F/ z [ y  /  x ] ph
43nfri 1972 1  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  hbab  2462  hblem  2579  bj-hblem  31526
  Copyright terms: Public domain W3C validator