Theorem rabsnt 3658

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 3614	. . 3 |- B e. { B }
3		id 19	. . 3 |- ( { x e. A | ph } = { B } -> { x e. A | ph } = { B } )
4	2, 3	eleqtrrid 2260	. 2 |- ( { x e. A | ph } = { B } -> B e. { x e. A | ph } )
5		rabsnt.2	. . . 4 |- ( x = B -> ( ph <-> ps ) )
6	5	elrab 2886	. . 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 1348   e. wcel 2141   {crab 2452   _Vcvv 2730   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-sn 3589

This theorem is referenced by:  ontr2exmid  4509  onsucsssucexmid  4511  ordsoexmid  4546  unfiexmid  6895