Theorem elab3 2836

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 2835	. 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 104   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  elrnmpo  5884  isfi  6655  addnqprlemfl  7367  addnqprlemfu  7368  mulnqprlemfl  7383  mulnqprlemfu  7384  istps  12199