Theorem rabn0m 3524

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 2517	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		rabid 2710	. . 3 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
3	2	exbii 1654	. 2 |- ( E. x x e. { x e. A | ph } <-> E. x ( x e. A /\ ph ) )
4		nfv 1577	. . 3 |- F/ y x e. { x e. A | ph }
5		df-rab 2520	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eleq2i 2298	. . . 4 |- ( y e. { x e. A | ph } <-> y e. { x | ( x e. A /\ ph ) } )
7		nfsab1 2221	. . . 4 |- F/ x y e. { x | ( x e. A /\ ph ) }
8	6, 7	nfxfr 1523	. . 3 |- F/ x y e. { x e. A | ph }
9		eleq1 2294	. . 3 |- ( x = y -> ( x e. { x e. A | ph } <-> y e. { x e. A | ph } ) )
10	4, 8, 9	cbvex 1804	. 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 1541   e. wcel 2202   {cab 2217   E.wrex 2512   {crab 2515

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-rex 2517  df-rab 2520

This theorem is referenced by:  exss  4325  cc4f  7548  cc4n  7550  nnwosdc  12690  lspf  14485  incistruhgr  16031