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Theorem ax10oe 1797
Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1715 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 1455 . . 3  |-  ( A. x  x  =  y  ->  A. x ( ps 
->  ( x  =  y  /\  ps ) ) )
3 exim 1599 . . 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 1796 . . 3  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps )
)
65sps 1537 . 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 1351    = wceq 1353   E.wex 1492
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-11 1506  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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