Theorem elab3 2959

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 2958	. 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 1398   e. wcel 2202   {cab 2217   _Vcvv 2803

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  elrnmpo  6145  isfi  6977  addnqprlemfl  7822  addnqprlemfu  7823  mulnqprlemfl  7838  mulnqprlemfu  7839  iswrd  11164  4sqlem2  13025  istps  14826  elply  15528