Theorem elab 2882

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 1528	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2881	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  Vcvv 2738

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740

This theorem is referenced by:  ralab  2898  rexab  2900  intab  3874  dfiin2g  3920  dfiunv2  3923  uniuni  4452  dcextest  4581  peano5  4598  finds  4600  finds2  4601  funcnvuni  5286  fun11iun  5483  elabrex  5759  abrexco  5760  mapval2  6678  ssenen  6851  snexxph  6949  sbthlem2  6957  indpi  7341  nqprm  7541  nqprrnd  7542  nqprdisj  7543  nqprloc  7544  nqprl  7550  nqpru  7551  cauappcvgprlem2  7659  caucvgprlem2  7679  peano1nnnn  7851  peano2nnnn  7852  1nn  8930  peano2nn  8931  dfuzi  9363  hashfacen  10816  shftfvalg  10827  ovshftex  10828  shftfval  10830  txdis1cn  13781  bj-ssom  14691