Theorem elab 2947

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 1574	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2946	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215  Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  ralab  2963  rexab  2965  intab  3952  dfiin2g  3998  dfiunv2  4001  uniuni  4543  dcextest  4674  peano5  4691  finds  4693  finds2  4694  funcnvuni  5393  fun11iun  5598  elabrex  5890  abrexco  5892  mapval2  6838  ssenen  7025  snexxph  7133  sbthlem2  7141  indpi  7545  nqprm  7745  nqprrnd  7746  nqprdisj  7747  nqprloc  7748  nqprl  7754  nqpru  7755  cauappcvgprlem2  7863  caucvgprlem2  7883  peano1nnnn  8055  peano2nnnn  8056  1nn  9137  peano2nn  9138  dfuzi  9573  hashfacen  11076  shftfvalg  11350  ovshftex  11351  shftfval  11353  4sqlemafi  12939  lss1d  14368  txdis1cn  14973  ushgredgedg  16045  ushgredgedgloop  16047  bj-ssom  16408