Theorem elab 2782

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 1476	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2781	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1299   ∈ wcel 1448  {cab 2086  Vcvv 2641

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643

This theorem is referenced by:  ralab  2797  rexab  2799  intab  3747  dfiin2g  3793  dfiunv2  3796  uniuni  4310  dcextest  4433  peano5  4450  finds  4452  finds2  4453  funcnvuni  5128  fun11iun  5322  elabrex  5591  abrexco  5592  mapval2  6502  ssenen  6674  snexxph  6766  sbthlem2  6774  indpi  7051  nqprm  7251  nqprrnd  7252  nqprdisj  7253  nqprloc  7254  nqprl  7260  nqpru  7261  cauappcvgprlem2  7369  caucvgprlem2  7389  peano1nnnn  7539  peano2nnnn  7540  1nn  8589  peano2nn  8590  dfuzi  9013  hashfacen  10420  shftfvalg  10431  ovshftex  10432  shftfval  10434  txdis1cn  12228  bj-ssom  12719