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Theorem ax10oe 1719
Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1644 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 106 . . . 4  |-  ( x  =  y  ->  ( ps  ->  ( x  =  y  /\  ps )
) )
21alimi 1385 . . 3  |-  ( A. x  x  =  y  ->  A. x ( ps 
->  ( x  =  y  /\  ps ) ) )
3 exim 1531 . . 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 1718 . . 3  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps )
)
65sps 1471 . 2  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ps )  ->  E. y ps ) )
74, 6syld 44 1  |-  ( A. x  x  =  y  ->  ( E. x ps 
->  E. y ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283    = wceq 1285   E.wex 1422
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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-11 1438  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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