Theorem elab 2960

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 1577	. 2 ⊢ Ⅎ𝑥𝜓
2		elab.1	. 2 ⊢ 𝐴 ∈ V
3		elab.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabf 2959	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {cab 2218  Vcvv 2812

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814

This theorem is referenced by:  ralab  2976  rexab  2978  intab  3977  dfiin2g  4023  dfiunv2  4026  uniuni  4571  dcextest  4702  peano5  4719  finds  4721  finds2  4722  funcnvuni  5424  fun11iun  5634  elabrex  5929  abrexco  5931  mapval2  6911  ssenen  7104  snexxph  7219  sbthlem2  7227  indpi  7653  nqprm  7853  nqprrnd  7854  nqprdisj  7855  nqprloc  7856  nqprl  7862  nqpru  7863  cauappcvgprlem2  7971  caucvgprlem2  7991  peano1nnnn  8163  peano2nnnn  8164  1nn  9244  peano2nn  9245  dfuzi  9684  hashfacen  11201  shftfvalg  11496  ovshftex  11497  shftfval  11499  4sqlemafi  13086  lss1d  14518  txdis1cn  15130  ushgredgedg  16208  ushgredgedgloop  16210  bj-ssom  16693