Theorem abn0m 3473

Description: Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)

Assertion

Ref	Expression
abn0m	|- ( E. y y e. { x | ph } <-> E. x ph )

Distinct variable groups:   ph, y   x, y

Allowed substitution hint:   ph( x)

Proof of Theorem abn0m

Step	Hyp	Ref	Expression
1		nfv 1539	. . 3 |- F/ y x e. { x | ph }
2		nfsab1 2183	. . 3 |- F/ x y e. { x | ph }
3		eleq1w 2254	. . 3 |- ( x = y -> ( x e. { x | ph } <-> y e. { x | ph } ) )
4	1, 2, 3	cbvex 1767	. 2 |- ( E. x x e. { x | ph } <-> E. y y e. { x | ph } )
5		abid 2181	. . 3 |- ( x e. { x | ph } <-> ph )
6	5	exbii 1616	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
7	4, 6	bitr3i 186	1 |- ( E. y y e. { x | ph } <-> E. x ph )

Syntax hints:   <-> wb 105   E.wex 1503   e. wcel 2164   {cab 2179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-clel 2189

This theorem is referenced by:  mapprc  6708