ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb6f Unicode version

Theorem sb6f 1700
Description: Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)
Hypothesis
Ref Expression
equs45f.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
sb6f  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)

Proof of Theorem sb6f
StepHypRef Expression
1 equs45f.1 . . . 4  |-  ( ph  ->  A. y ph )
21sbimi 1663 . . 3  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] A. y ph )
3 sb4a 1698 . . 3  |-  ( [ y  /  x ] A. y ph  ->  A. x
( x  =  y  ->  ph ) )
42, 3syl 14 . 2  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)
5 sb2 1666 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
64, 5impbii 121 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-11 1413  ax-4 1416  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sb5f  1701  sbcof2  1707
  Copyright terms: Public domain W3C validator