Theorem elab 2970

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)

Hypotheses

Ref	Expression
elab.1	|- A e. _V
elab.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elab	|- ( A e. { x | ph } <-> ps )

Distinct variable groups:   ps, x   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem elab

Step	Hyp	Ref	Expression
1		nfv 1581	. 2 |- F/ x ps
2		elab.1	. 2 |- A e. _V
3		elab.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabf 2969	1 |- ( A e. { x | ph } <-> 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:  ralab  2986  rexab  2988  intab  3999  dfiin2g  4045  dfiunv2  4048  uniuni  4599  dcextest  4730  peano5  4747  finds  4749  finds2  4750  funcnvuni  5453  fun11iun  5663  elabrex  5966  abrexco  5968  mapfset  6949  mapfoss  6951  fsetsspwxp  6952  mapval2  6963  ssenen  7156  snexxph  7272  sbthlem2  7280  f1setfi  7322  indpi  7715  nqprm  7915  nqprrnd  7916  nqprdisj  7917  nqprloc  7918  nqprl  7924  nqpru  7925  cauappcvgprlem2  8033  caucvgprlem2  8053  peano1nnnn  8225  peano2nnnn  8226  1nn  9324  peano2nn  9325  dfuzi  9767  hashfacen  11306  hashf1lem1  11307  hashf1lem2  11308  shftfvalg  11605  ovshftex  11606  shftfval  11608  4sqlemafi  13203  lss1d  14829  txdis1cn  15504  ushgredgedg  16672  ushgredgedgloop  16674  bj-ssom  17167