Theorem rabsnt 3741

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 3697	. . 3 |- B e. { B }
3		id 19	. . 3 |- ( { x e. A | ph } = { B } -> { x e. A | ph } = { B } )
4	2, 3	eleqtrrid 2319	. 2 |- ( { x e. A | ph } = { B } -> B e. { x e. A | ph } )
5		rabsnt.2	. . . 4 |- ( x = B -> ( ph <-> ps ) )
6	5	elrab 2959	. . 3 |- ( B e. { x e. A | ph } <-> ( B e. A /\ ps ) )
7	6	simprbi 275	. 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 105   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   {csn 3666

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-sn 3672

This theorem is referenced by:  ontr2exmid  4617  onsucsssucexmid  4619  ordsoexmid  4654  unfiexmid  7080