ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax10oe Unicode version

Theorem ax10oe 1807
Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1725 but for  E. rather than  A.. (Contributed by Jim Kingdon, 21-Dec-2017.)
Assertion
Ref Expression
ax10oe  |-  ( A. x  x  =  y  ->  ( E. x ps 
->  E. y ps )
)

Proof of Theorem ax10oe
StepHypRef Expression
1 ax-ia3 108 . . . 4  |-  ( x  =  y  ->  ( ps  ->  ( x  =  y  /\  ps )
) )
21alimi 1465 . . 3  |-  ( A. x  x  =  y  ->  A. x ( ps 
->  ( x  =  y  /\  ps ) ) )
3 exim 1609 . . 3  |-  ( A. x ( ps  ->  ( x  =  y  /\  ps ) )  ->  ( E. x ps  ->  E. x
( x  =  y  /\  ps ) ) )
42, 3syl 14 . 2  |-  ( A. x  x  =  y  ->  ( E. x ps 
->  E. x ( x  =  y  /\  ps ) ) )
5 ax11e 1806 . . 3  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps )
)
65sps 1547 . 2  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps ) )
74, 6syld 45 1  |-  ( A. x  x  =  y  ->  ( E. x ps 
->  E. y ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1361    = wceq 1363   E.wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-11 1516  ax-4 1520  ax-ial 1544
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator