Theorem rabn0m 3478

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 2481	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		rabid 2673	. . 3 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
3	2	exbii 1619	. 2 |- ( E. x x e. { x e. A | ph } <-> E. x ( x e. A /\ ph ) )
4		nfv 1542	. . 3 |- F/ y x e. { x e. A | ph }
5		df-rab 2484	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eleq2i 2263	. . . 4 |- ( y e. { x e. A | ph } <-> y e. { x | ( x e. A /\ ph ) } )
7		nfsab1 2186	. . . 4 |- F/ x y e. { x | ( x e. A /\ ph ) }
8	6, 7	nfxfr 1488	. . 3 |- F/ x y e. { x e. A | ph }
9		eleq1 2259	. . 3 |- ( x = y -> ( x e. { x e. A | ph } <-> y e. { x e. A | ph } ) )
10	4, 8, 9	cbvex 1770	. 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 1506   e. wcel 2167   {cab 2182   E.wrex 2476   {crab 2479

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rex 2481  df-rab 2484

This theorem is referenced by:  exss  4260  cc4f  7336  cc4n  7338  nnwosdc  12206  lspf  13945