Theorem elab 2823

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 1508	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2822	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2123  Vcvv 2681

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  ralab  2839  rexab  2841  intab  3795  dfiin2g  3841  dfiunv2  3844  uniuni  4367  dcextest  4490  peano5  4507  finds  4509  finds2  4510  funcnvuni  5187  fun11iun  5381  elabrex  5652  abrexco  5653  mapval2  6565  ssenen  6738  snexxph  6831  sbthlem2  6839  indpi  7143  nqprm  7343  nqprrnd  7344  nqprdisj  7345  nqprloc  7346  nqprl  7352  nqpru  7353  cauappcvgprlem2  7461  caucvgprlem2  7481  peano1nnnn  7653  peano2nnnn  7654  1nn  8724  peano2nn  8725  dfuzi  9154  hashfacen  10572  shftfvalg  10583  ovshftex  10584  shftfval  10586  txdis1cn  12436  bj-ssom  13123