Theorem elab 2870

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 1516	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2869	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {cab 2151  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  ralab  2886  rexab  2888  intab  3853  dfiin2g  3899  dfiunv2  3902  uniuni  4429  dcextest  4558  peano5  4575  finds  4577  finds2  4578  funcnvuni  5257  fun11iun  5453  elabrex  5726  abrexco  5727  mapval2  6644  ssenen  6817  snexxph  6915  sbthlem2  6923  indpi  7283  nqprm  7483  nqprrnd  7484  nqprdisj  7485  nqprloc  7486  nqprl  7492  nqpru  7493  cauappcvgprlem2  7601  caucvgprlem2  7621  peano1nnnn  7793  peano2nnnn  7794  1nn  8868  peano2nn  8869  dfuzi  9301  hashfacen  10749  shftfvalg  10760  ovshftex  10761  shftfval  10763  txdis1cn  12918  bj-ssom  13818