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Theorem bj-ex 10833
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 1530 and 19.9ht 1573 or 19.23ht 1427). (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 1424 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x (
ph  ->  ph )  <->  ( E. x ph  ->  ph ) ) )
2 ax-17 1460 . . 3  |-  ( ph  ->  A. x ph )
31, 2mpg 1381 . 2  |-  ( A. x ( ph  ->  ph )  <->  ( E. x ph  ->  ph ) )
4 id 19 . 2  |-  ( ph  ->  ph )
53, 4mpgbi 1382 1  |-  ( E. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-gen 1379  ax-ie2 1424  ax-17 1460
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bj-d0clsepcl  10987  bj-inf2vnlem1  11032  bj-nn0sucALT  11040
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