Theorem bj-ex 13140

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 1578 and 19.9ht 1621 or 19.23ht 1474). (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ex	|- ( E. x ph -> ph )

Distinct variable group:   ph, x

Proof of Theorem bj-ex

Step	Hyp	Ref	Expression
1		ax-ie2 1471	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( A. x ( ph -> ph ) <-> ( E. x ph -> ph ) ) )
2		ax-17 1507	. . 3 |- ( ph -> A. x ph )
3	1, 2	mpg 1428	. 2 |- ( A. x ( ph -> ph ) <-> ( E. x ph -> ph ) )
4		id 19	. 2 |- ( ph -> ph )
5	3, 4	mpgbi 1429	1 |- ( E. x ph -> ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   E.wex 1469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1426  ax-ie2 1471  ax-17 1507

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bj-d0clsepcl  13294  bj-inf2vnlem1  13339  bj-nn0sucALT  13347