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Theorem bj-ex 13718
Description: Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1591 and 19.9ht 1634 or 19.23ht 1490). (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ex  |-  ( E. x ph  ->  ph )
Distinct variable group:    ph, x

Proof of Theorem bj-ex
StepHypRef Expression
1 ax-ie2 1487 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x (
ph  ->  ph )  <->  ( E. x ph  ->  ph ) ) )
2 ax-17 1519 . . 3  |-  ( ph  ->  A. x ph )
31, 2mpg 1444 . 2  |-  ( A. x ( ph  ->  ph )  <->  ( E. x ph  ->  ph ) )
4 id 19 . 2  |-  ( ph  ->  ph )
53, 4mpgbi 1445 1  |-  ( E. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1346   E.wex 1485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1442  ax-ie2 1487  ax-17 1519
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  bj-d0clsepcl  13882  bj-inf2vnlem1  13927  bj-nn0sucALT  13935
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