Theorem elab 2963

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 ∈ V
elab.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elab	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elab

Step	Hyp	Ref	Expression
1		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2962	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {cab 2220  Vcvv 2815

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817

This theorem is referenced by:  ralab  2979  rexab  2981  intab  3980  dfiin2g  4026  dfiunv2  4029  uniuni  4574  dcextest  4705  peano5  4722  finds  4724  finds2  4725  funcnvuni  5427  fun11iun  5637  elabrex  5932  abrexco  5934  mapval2  6914  ssenen  7107  snexxph  7222  sbthlem2  7230  indpi  7662  nqprm  7862  nqprrnd  7863  nqprdisj  7864  nqprloc  7865  nqprl  7871  nqpru  7872  cauappcvgprlem2  7980  caucvgprlem2  8000  peano1nnnn  8172  peano2nnnn  8173  1nn  9253  peano2nn  9254  dfuzi  9694  hashfacen  11216  shftfvalg  11511  ovshftex  11512  shftfval  11514  4sqlemafi  13101  lss1d  14580  txdis1cn  15192  ushgredgedg  16270  ushgredgedgloop  16272  bj-ssom  16755