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Theorem bj-ex 11618
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 1534 and 19.9ht 1577 or 19.23ht 1431). (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 1428 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x (
ph  ->  ph )  <->  ( E. x ph  ->  ph ) ) )
2 ax-17 1464 . . 3  |-  ( ph  ->  A. x ph )
31, 2mpg 1385 . 2  |-  ( A. x ( ph  ->  ph )  <->  ( E. x ph  ->  ph ) )
4 id 19 . 2  |-  ( ph  ->  ph )
53, 4mpgbi 1386 1  |-  ( E. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-gen 1383  ax-ie2 1428  ax-17 1464
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bj-d0clsepcl  11775  bj-inf2vnlem1  11820  bj-nn0sucALT  11828
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