ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb4bor Unicode version

Theorem sb4bor 1758
Description: Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
Assertion
Ref Expression
sb4bor  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )

Proof of Theorem sb4bor
StepHypRef Expression
1 sb4or 1756 . 2  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 1692 . . . . 5  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
3 df-bi 115 . . . . . 6  |-  ( ( ( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) )  -> 
( ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)  /\  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph ) ) )  /\  ( ( ( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) )  /\  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph ) )  -> 
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) ) )
43simpri 111 . . . . 5  |-  ( ( ( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) )  /\  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph ) )  -> 
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) )
52, 4mpan2 416 . . . 4  |-  ( ( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) )  -> 
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) )
65alimi 1385 . . 3  |-  ( A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)  ->  A. x
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) )
76orim2i 711 . 2  |-  ( ( A. x  x  =  y  \/  A. x
( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) ) )  ->  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph ) ) ) )
81, 7ax-mp 7 1  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662   A.wal 1283   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator