Theorem elab3 2924

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 2923	. 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 1372   e. wcel 2175   {cab 2190   _Vcvv 2771

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773

This theorem is referenced by:  elrnmpo  6049  isfi  6838  addnqprlemfl  7654  addnqprlemfu  7655  mulnqprlemfl  7670  mulnqprlemfu  7671  iswrd  10971  4sqlem2  12631  istps  14422  elply  15124