Theorem elab3 2904

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 2903	. 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 1364   e. wcel 2160   {cab 2175   _Vcvv 2752

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  elrnmpo  6009  isfi  6786  addnqprlemfl  7587  addnqprlemfu  7588  mulnqprlemfl  7603  mulnqprlemfu  7604  4sqlem2  12420  istps  13984