Theorem elab 2904

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 1539	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2903	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  Vcvv 2760

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  ralab  2920  rexab  2922  intab  3899  dfiin2g  3945  dfiunv2  3948  uniuni  4482  dcextest  4613  peano5  4630  finds  4632  finds2  4633  funcnvuni  5323  fun11iun  5521  elabrex  5800  abrexco  5802  mapval2  6732  ssenen  6907  snexxph  7009  sbthlem2  7017  indpi  7402  nqprm  7602  nqprrnd  7603  nqprdisj  7604  nqprloc  7605  nqprl  7611  nqpru  7612  cauappcvgprlem2  7720  caucvgprlem2  7740  peano1nnnn  7912  peano2nnnn  7913  1nn  8993  peano2nn  8994  dfuzi  9427  hashfacen  10907  shftfvalg  10962  ovshftex  10963  shftfval  10965  4sqlemafi  12533  lss1d  13879  txdis1cn  14446  bj-ssom  15428