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Theorem bj-elissetv 31381
Description: Version of bj-elisset 31382 with a dv condition on  x ,  V. This proof uses only df-ex 1658, ax-gen 1663, ax-4 1676 and df-clel 2424 on top of propositional calculus. Prefer its use over bj-elisset 31382 when sufficient. (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-elissetv  |-  ( A  e.  V  ->  E. x  x  =  A )
Distinct variable groups:    x, A    x, V

Proof of Theorem bj-elissetv
StepHypRef Expression
1 df-clel 2424 . 2  |-  ( A  e.  V  <->  E. x
( x  =  A  /\  x  e.  V
) )
2 exsimpl 1723 . 2  |-  ( E. x ( x  =  A  /\  x  e.  V )  ->  E. x  x  =  A )
31, 2sylbi 198 1  |-  ( A  e.  V  ->  E. x  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-clel 2424
This theorem is referenced by:  bj-elisset  31382  bj-issetiv  31383  bj-ceqsaltv  31396  bj-ceqsalgv  31400  bj-vtoclg1fv  31430  bj-ru  31450
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