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Theorem ax11ev 1782
Description: Analogue to ax11v 1781 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
Assertion
Ref Expression
ax11ev  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax11ev
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 a9e 1657 . 2  |-  E. z 
z  =  y
2 ax11e 1750 . . . . 5  |-  ( x  =  z  ->  ( E. x ( x  =  z  /\  ph )  ->  E. z ph )
)
3 ax-17 1489 . . . . . 6  |-  ( ph  ->  A. z ph )
4319.9h 1605 . . . . 5  |-  ( E. z ph  <->  ph )
52, 4syl6ib 160 . . . 4  |-  ( x  =  z  ->  ( E. x ( x  =  z  /\  ph )  ->  ph ) )
6 equequ2 1672 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
76anbi1d 458 . . . . . . 7  |-  ( z  =  y  ->  (
( x  =  z  /\  ph )  <->  ( x  =  y  /\  ph )
) )
87exbidv 1779 . . . . . 6  |-  ( z  =  y  ->  ( E. x ( x  =  z  /\  ph )  <->  E. x ( x  =  y  /\  ph )
) )
98imbi1d 230 . . . . 5  |-  ( z  =  y  ->  (
( E. x ( x  =  z  /\  ph )  ->  ph )  <->  ( E. x ( x  =  y  /\  ph )  ->  ph ) ) )
106, 9imbi12d 233 . . . 4  |-  ( z  =  y  ->  (
( x  =  z  ->  ( E. x
( x  =  z  /\  ph )  ->  ph ) )  <->  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) ) ) )
115, 10mpbii 147 . . 3  |-  ( z  =  y  ->  (
x  =  y  -> 
( E. x ( x  =  y  /\  ph )  ->  ph ) ) )
1211exlimiv 1560 . 2  |-  ( E. z  z  =  y  ->  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) ) )
131, 12ax-mp 5 1  |-  ( x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   E.wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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