Theorem elab2 2931

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Hypotheses

Ref	Expression
elab2.1	⊢ 𝐴 ∈ V
elab2.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
elab2.3	⊢ 𝐵 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
elab2	⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elab2

Step	Hyp	Ref	Expression
1		elab2.1	. 2 ⊢ 𝐴 ∈ V
2		elab2.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3		elab2.3	. . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑}
4	2, 3	elab2g 2930	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1375   ∈ wcel 2180  {cab 2195  Vcvv 2779

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781

This theorem is referenced by:  elpw  3635  elint  3908  opabid  4323  elrn2  4942  elimasn  5071  oprabid  6006  tfrlem3a  6426  tfrcllemsucaccv  6470  tfrcllembxssdm  6472  tfrcllemres  6478  addnqprlemrl  7712  addnqprlemru  7713  addnqprlemfl  7714  addnqprlemfu  7715  mulnqprlemrl  7728  mulnqprlemru  7729  mulnqprlemfl  7730  mulnqprlemfu  7731  ltnqpr  7748  ltnqpri  7749  archpr  7798  cauappcvgprlemladdfu  7809  cauappcvgprlemladdfl  7810  caucvgprlemladdfu  7832  caucvgprprlemopu  7854  suplocexprlemloc  7876  4sqlem12  12891  txuni2  14895