Theorem abn0m 3388

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 1508	. . 3 |- F/ y x e. { x | ph }
2		nfsab1 2129	. . 3 |- F/ x y e. { x | ph }
3		eleq1w 2200	. . 3 |- ( x = y -> ( x e. { x | ph } <-> y e. { x | ph } ) )
4	1, 2, 3	cbvex 1729	. 2 |- ( E. x x e. { x | ph } <-> E. y y e. { x | ph } )
5		abid 2127	. . 3 |- ( x e. { x | ph } <-> ph )
6	5	exbii 1584	. 2 |- ( E. x x e. { x | ph } <-> E. x ph )
7	4, 6	bitr3i 185	1 |- ( E. y y e. { x | ph } <-> E. x ph )

Syntax hints:   <-> wb 104   E.wex 1468   e. wcel 1480   {cab 2125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-clel 2135

This theorem is referenced by:  mapprc  6546