ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbh Unicode version

Theorem sbh 1732
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)
Hypothesis
Ref Expression
sbh.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
sbh  |-  ( [ y  /  x ] ph 
<-> 
ph )

Proof of Theorem sbh
StepHypRef Expression
1 sb1 1722 . . . 4  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 sbh.1 . . . . 5  |-  ( ph  ->  A. x ph )
3219.41h 1646 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  <->  ( E. x  x  =  y  /\  ph )
)
41, 3sylib 121 . . 3  |-  ( [ y  /  x ] ph  ->  ( E. x  x  =  y  /\  ph ) )
54simprd 113 . 2  |-  ( [ y  /  x ] ph  ->  ph )
6 stdpc4 1731 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
72, 6syl 14 . 2  |-  ( ph  ->  [ y  /  x ] ph )
85, 7impbii 125 1  |-  ( [ y  /  x ] ph 
<-> 
ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312   E.wex 1451   [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-sb 1719
This theorem is referenced by:  sbf  1733  sb6x  1735  nfs1f  1736  hbs1f  1737  sbid2h  1803  sblimv  1848  sbrim  1905  sbrbif  1911  elsb3  1927  elsb4  1928
  Copyright terms: Public domain W3C validator