Theorem rabn0m 3448

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 2459	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		rabid 2650	. . 3 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
3	2	exbii 1603	. 2 |- ( E. x x e. { x e. A | ph } <-> E. x ( x e. A /\ ph ) )
4		nfv 1526	. . 3 |- F/ y x e. { x e. A | ph }
5		df-rab 2462	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eleq2i 2242	. . . 4 |- ( y e. { x e. A | ph } <-> y e. { x | ( x e. A /\ ph ) } )
7		nfsab1 2165	. . . 4 |- F/ x y e. { x | ( x e. A /\ ph ) }
8	6, 7	nfxfr 1472	. . 3 |- F/ x y e. { x e. A | ph }
9		eleq1 2238	. . 3 |- ( x = y -> ( x e. { x e. A | ph } <-> y e. { x e. A | ph } ) )
10	4, 8, 9	cbvex 1754	. 2 |- ( E. x x e. { x e. A | ph } <-> E. y y e. { x e. A | ph } )
11	1, 3, 10	3bitr2ri 209	1 |- ( E. y y e. { x e. A | ph } <-> E. x e. A ph )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1490   e. wcel 2146   {cab 2161   E.wrex 2454   {crab 2457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-rex 2459  df-rab 2462

This theorem is referenced by:  exss  4221  cc4f  7243  cc4n  7245  nnwosdc  12007