Theorem rabn0m 3436

Description: Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)

Assertion

Ref	Expression
rabn0m	|- ( E. y y e. { x e. A | ph } <-> E. x e. A ph )

Distinct variable groups:   x, y   y, A   ph, y

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem rabn0m

Step	Hyp	Ref	Expression
1		df-rex 2450	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		rabid 2641	. . 3 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
3	2	exbii 1593	. 2 |- ( E. x x e. { x e. A | ph } <-> E. x ( x e. A /\ ph ) )
4		nfv 1516	. . 3 |- F/ y x e. { x e. A | ph }
5		df-rab 2453	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eleq2i 2233	. . . 4 |- ( y e. { x e. A | ph } <-> y e. { x | ( x e. A /\ ph ) } )
7		nfsab1 2155	. . . 4 |- F/ x y e. { x | ( x e. A /\ ph ) }
8	6, 7	nfxfr 1462	. . 3 |- F/ x y e. { x e. A | ph }
9		eleq1 2229	. . 3 |- ( x = y -> ( x e. { x e. A | ph } <-> y e. { x e. A | ph } ) )
10	4, 8, 9	cbvex 1744	. 2 |- ( E. x x e. { x e. A | ph } <-> E. y y e. { x e. A | ph } )
11	1, 3, 10	3bitr2ri 208	1 |- ( E. y y e. { x e. A | ph } <-> E. x e. A ph )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1480   e. wcel 2136   {cab 2151   E.wrex 2445   {crab 2448

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  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-rex 2450  df-rab 2453

This theorem is referenced by:  exss  4205  cc4f  7210  cc4n  7212  nnwosdc  11972