Theorem bj-issetwt 32859

Description: Closed form of bj-issetw 32860. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-issetwt	|- ( A. x ph -> ( A e. { x | ph } <-> E. y y = A ) )

Distinct variable group:   y, A

Allowed substitution hints:   ph( x, y)   A( x)

Proof of Theorem bj-issetwt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2618	. . 3 |- ( A e. { x | ph } <-> E. z ( z = A /\ z e. { x | ph } ) )
2	1	a1i 11	. 2 |- ( A. x ph -> ( A e. { x | ph } <-> E. z ( z = A /\ z e. { x | ph } ) ) )
3		bj-vexwvt 32856	. . . . 5 |- ( A. x ph -> z e. { x | ph } )
4	3	biantrud 528	. . . 4 |- ( A. x ph -> ( z = A <-> ( z = A /\ z e. { x | ph } ) ) )
5	4	bicomd 213	. . 3 |- ( A. x ph -> ( ( z = A /\ z e. { x | ph } ) <-> z = A ) )
6	5	exbidv 1850	. 2 |- ( A. x ph -> ( E. z ( z = A /\ z e. { x | ph } ) <-> E. z z = A ) )
7		bj-denotes 32858	. . 3 |- ( E. z z = A <-> E. y y = A )
8	7	a1i 11	. 2 |- ( A. x ph -> ( E. z z = A <-> E. y y = A ) )
9	2, 6, 8	3bitrd 294	1 |- ( A. x ph -> ( A e. { x | ph } <-> E. y y = A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608

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-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-clel 2618

This theorem is referenced by:  bj-issetw  32860