Theorem bj-cleljusti 32669

Description: One direction of cleljust 1998, requiring only ax-1 6-- ax-5 1839 and ax8v1 1994. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cleljusti	|- ( E. z ( z = x /\ z e. y ) -> x e. y )

Distinct variable groups:   x, z   y, z

Proof of Theorem bj-cleljusti

Step	Hyp	Ref	Expression
1		ax8v1 1994	. . 3 |- ( z = x -> ( z e. y -> x e. y ) )
2	1	imp 445	. 2 |- ( ( z = x /\ z e. y ) -> x e. y )
3	2	exlimiv 1858	1 |- ( E. z ( z = x /\ z e. y ) -> x e. y )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-8 1992

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)