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Theorem ax11e 1719
Description: Analogue to ax-11 1438 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)
Assertion
Ref Expression
ax11e  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  E. y ph )
)

Proof of Theorem ax11e
StepHypRef Expression
1 equs5e 1718 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
2119.21bi 1491 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ( x  =  y  ->  E. y ph )
)
32com12 30 1  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  E. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = 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:  ax10oe  1720  drex1  1721  sbcof2  1733  ax11ev  1751
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