Theorem abn0m 3547

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 1581	. . 3 |- F/ y x e. { x | ph }
2		nfsab1 2228	. . 3 |- F/ x y e. { x | ph }
3		eleq1w 2299	. . 3 |- ( x = y -> ( x e. { x | ph } <-> y e. { x | ph } ) )
4	1, 2, 3	cbvex 1809	. 2 |- ( E. x x e. { x | ph } <-> E. y y e. { x | ph } )
5		abid 2226	. . 3 |- ( x e. { x | ph } <-> ph )
6	5	exbii 1658	. 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 )

This proof depends on syntax axioms:   <-> wb 105   E.wex 1545   e. wcel 2209   {cab 2224

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-clel 2234

This theorem is used by:  mapprc  6926  acnrcl  7557  hashf1lem2  11286  hashf1  11287  birthdaylem3  16089