Theorem abn0m 3434

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 1516	. . 3 |- F/ y x e. { x | ph }
2		nfsab1 2155	. . 3 |- F/ x y e. { x | ph }
3		eleq1w 2227	. . 3 |- ( x = y -> ( x e. { x | ph } <-> y e. { x | ph } ) )
4	1, 2, 3	cbvex 1744	. 2 |- ( E. x x e. { x | ph } <-> E. y y e. { x | ph } )
5		abid 2153	. . 3 |- ( x e. { x | ph } <-> ph )
6	5	exbii 1593	. 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 1480   e. wcel 2136   {cab 2151

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-clel 2161

This theorem is referenced by:  mapprc  6618