Theorem rabn0m 3245

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 2312	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		rabid 2485	. . 3 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
3	2	exbii 1496	. 2 |- ( E. x x e. { x e. A | ph } <-> E. x ( x e. A /\ ph ) )
4		nfv 1421	. . 3 |- F/ y x e. { x e. A | ph }
5		df-rab 2315	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eleq2i 2104	. . . 4 |- ( y e. { x e. A | ph } <-> y e. { x | ( x e. A /\ ph ) } )
7		nfsab1 2030	. . . 4 |- F/ x y e. { x | ( x e. A /\ ph ) }
8	6, 7	nfxfr 1363	. . 3 |- F/ x y e. { x e. A | ph }
9		eleq1 2100	. . 3 |- ( x = y -> ( x e. { x e. A | ph } <-> y e. { x e. A | ph } ) )
10	4, 8, 9	cbvex 1639	. 2 |- ( E. x x e. { x e. A | ph } <-> E. y y e. { x e. A | ph } )
11	1, 3, 10	3bitr2ri 198	1 |- ( E. y y e. { x e. A | ph } <-> E. x e. A ph )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   {cab 2026   E.wrex 2307   {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rex 2312  df-rab 2315

This theorem is referenced by:  exss  3963