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Theorem bj-elissetv 34838
Description: Version of bj-elisset 34839 with a dv condition on  x ,  V. This proof uses only df-ex 1618, ax-gen 1623, ax-4 1636 and df-clel 2449 on top of propositional calculus. Prefer its use over bj-elisset 34839 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 2449 . 2  |-  ( A  e.  V  <->  E. x
( x  =  A  /\  x  e.  V
) )
2 exsimpl 1682 . 2  |-  ( E. x ( x  =  A  /\  x  e.  V )  ->  E. x  x  =  A )
31, 2sylbi 195 1  |-  ( A  e.  V  ->  E. x  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-clel 2449
This theorem is referenced by:  bj-elisset  34839  bj-issetiv  34840  bj-ceqsaltv  34853  bj-ceqsalgv  34857  bj-vtoclg1fv  34885  bj-ru  34903
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