Theorem rabsnt 3651

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 3607	. . 3 |- B e. { B }
3		id 19	. . 3 |- ( { x e. A | ph } = { B } -> { x e. A | ph } = { B } )
4	2, 3	eleqtrrid 2256	. 2 |- ( { x e. A | ph } = { B } -> B e. { x e. A | ph } )
5		rabsnt.2	. . . 4 |- ( x = B -> ( ph <-> ps ) )
6	5	elrab 2882	. . 3 |- ( B e. { x e. A | ph } <-> ( B e. A /\ ps ) )
7	6	simprbi 273	. 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 104   = wceq 1343   e. wcel 2136   {crab 2448   _Vcvv 2726   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-sn 3582

This theorem is referenced by:  ontr2exmid  4502  onsucsssucexmid  4504  ordsoexmid  4539  unfiexmid  6883