Theorem elab 2964

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 2963	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  2980  rexab  2982  intab  3983  dfiin2g  4029  dfiunv2  4032  uniuni  4577  dcextest  4708  peano5  4725  finds  4727  finds2  4728  funcnvuni  5430  fun11iun  5640  elabrex  5936  abrexco  5938  mapval2  6925  ssenen  7118  snexxph  7233  sbthlem2  7241  indpi  7673  nqprm  7873  nqprrnd  7874  nqprdisj  7875  nqprloc  7876  nqprl  7882  nqpru  7883  cauappcvgprlem2  7991  caucvgprlem2  8011  peano1nnnn  8183  peano2nnnn  8184  1nn  9268  peano2nn  9269  dfuzi  9709  hashfacen  11236  shftfvalg  11531  ovshftex  11532  shftfval  11534  4sqlemafi  13121  lss1d  14660  txdis1cn  15272  ushgredgedg  16350  ushgredgedgloop  16352  bj-ssom  16845