Theorem bj-issetw 32860

Description: The closest one can get to isset 3207 without using ax-ext 2602. See also bj-vexw 32855. Note that the only dv condition is between y and A. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-issetw.1	|- ph

Assertion

Ref	Expression
bj-issetw	|- ( 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-issetw

Step	Hyp	Ref	Expression
1		bj-issetwt 32859	. 2 |- ( A. x ph -> ( A e. { x | ph } <-> E. y y = A ) )
2		bj-issetw.1	. 2 |- ph
3	1, 2	mpg 1724	1 |- ( A e. { x | ph } <-> E. y y = A )

Syntax hints:   <-> wb 196   = 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: (None)