Theorem elab2 2974

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Hypotheses

Ref	Expression
elab2.1	|- A e. _V
elab2.2	|- ( x = A -> ( ph <-> ps ) )
elab2.3	|- B = { x | ph }

Assertion

Ref	Expression
elab2	|- ( A e. B <-> ps )

Distinct variable groups:   ps, x   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem elab2

Step	Hyp	Ref	Expression
1		elab2.1	. 2 |- A e. _V
2		elab2.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3		elab2.3	. . 3 |- B = { x | ph }
4	2, 3	elab2g 2973	. 2 |- ( A e. _V -> ( A e. B <-> ps ) )
5	1, 4	ax-mp 5	1 |- ( A e. B <-> ps )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is used by:  elpw  3694  elint  3976  opabid  4398  elrn2  5024  elimasn  5154  oprabid  6117  tfrlem3a  6581  tfrcllemsucaccv  6625  tfrcllembxssdm  6627  tfrcllemres  6633  addnqprlemrl  7925  addnqprlemru  7926  addnqprlemfl  7927  addnqprlemfu  7928  mulnqprlemrl  7941  mulnqprlemru  7942  mulnqprlemfl  7943  mulnqprlemfu  7944  ltnqpr  7961  ltnqpri  7962  archpr  8011  cauappcvgprlemladdfu  8022  cauappcvgprlemladdfl  8023  caucvgprlemladdfu  8045  caucvgprprlemopu  8067  suplocexprlemloc  8089  4sqlem12  13204  txuni2  15448