Theorem elab3 2929

Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

Hypotheses

Ref	Expression
elab3.1	|- ( ps -> A e. _V )
elab3.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elab3	|- ( A e. { x | ph } <-> ps )

Distinct variable groups:   ps, x   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem elab3

Step	Hyp	Ref	Expression
1		elab3.1	. 2 |- ( ps -> A e. _V )
2		elab3.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	elab3g 2928	. 2 |- ( ( ps -> A e. _V ) -> ( A e. { x | ph } <-> ps ) )
4	1, 3	ax-mp 5	1 |- ( A e. { x | ph } <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2177   {cab 2192   _Vcvv 2773

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775

This theorem is referenced by:  elrnmpo  6072  isfi  6865  addnqprlemfl  7692  addnqprlemfu  7693  mulnqprlemfl  7708  mulnqprlemfu  7709  iswrd  11018  4sqlem2  12787  istps  14579  elply  15281