Theorem elab 2905

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 2904	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  2921  rexab  2923  intab  3900  dfiin2g  3946  dfiunv2  3949  uniuni  4483  dcextest  4614  peano5  4631  finds  4633  finds2  4634  funcnvuni  5324  fun11iun  5522  elabrex  5801  abrexco  5803  mapval2  6734  ssenen  6909  snexxph  7011  sbthlem2  7019  indpi  7404  nqprm  7604  nqprrnd  7605  nqprdisj  7606  nqprloc  7607  nqprl  7613  nqpru  7614  cauappcvgprlem2  7722  caucvgprlem2  7742  peano1nnnn  7914  peano2nnnn  7915  1nn  8995  peano2nn  8996  dfuzi  9430  hashfacen  10910  shftfvalg  10965  ovshftex  10966  shftfval  10968  4sqlemafi  12536  lss1d  13882  txdis1cn  14457  bj-ssom  15498