Theorem rabsnt 3467

Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
rabsnt.1	|- B e. _V
rabsnt.2	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
rabsnt	|- ( { x e. A | ph } = { B } -> ps )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem rabsnt

Step	Hyp	Ref	Expression
1		rabsnt.1	. . . 4 |- B e. _V
2	1	snid 3425	. . 3 |- B e. { B }
3		id 19	. . 3 |- ( { x e. A | ph } = { B } -> { x e. A | ph } = { B } )
4	2, 3	syl5eleqr 2168	. 2 |- ( { x e. A | ph } = { B } -> B e. { x e. A | ph } )
5		rabsnt.2	. . . 4 |- ( x = B -> ( ph <-> ps ) )
6	5	elrab 2749	. . 3 |- ( B e. { x e. A | ph } <-> ( B e. A /\ ps ) )
7	6	simprbi 269	. 2 |- ( B e. { x e. A | ph } -> ps )
8	4, 7	syl 14	1 |- ( { x e. A | ph } = { B } -> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   {crab 2352   _Vcvv 2601   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-sn 3404

This theorem is referenced by:  ontr2exmid  4268  onsucsssucexmid  4270  ordsoexmid  4305