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 ∈ 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 1581	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2969	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402   ∈ 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  4597  dcextest  4728  peano5  4745  finds  4747  finds2  4748  funcnvuni  5450  fun11iun  5660  elabrex  5963  abrexco  5965  mapfset  6945  mapfoss  6947  fsetsspwxp  6948  mapval2  6959  ssenen  7152  snexxph  7267  sbthlem2  7275  f1setfi  7317  indpi  7709  nqprm  7909  nqprrnd  7910  nqprdisj  7911  nqprloc  7912  nqprl  7918  nqpru  7919  cauappcvgprlem2  8027  caucvgprlem2  8047  peano1nnnn  8219  peano2nnnn  8220  1nn  9316  peano2nn  9317  dfuzi  9758  hashfacen  11286  hashf1lem1  11287  hashf1lem2  11288  shftfvalg  11585  ovshftex  11586  shftfval  11588  4sqlemafi  13176  lss1d  14722  txdis1cn  15381  ushgredgedg  16479  ushgredgedgloop  16481  bj-ssom  16974