Theorem abn0m 3477

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 1542	. . 3 |- F/ y x e. { x | ph }
2		nfsab1 2186	. . 3 |- F/ x y e. { x | ph }
3		eleq1w 2257	. . 3 |- ( x = y -> ( x e. { x | ph } <-> y e. { x | ph } ) )
4	1, 2, 3	cbvex 1770	. 2 |- ( E. x x e. { x | ph } <-> E. y y e. { x | ph } )
5		abid 2184	. . 3 |- ( x e. { x | ph } <-> ph )
6	5	exbii 1619	. 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 1506   e. wcel 2167   {cab 2182

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-clel 2192

This theorem is referenced by:  mapprc  6720  acnrcl  7284