Theorem elab2 2751

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 2750	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓))
5	1, 4	ax-mp 7	1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1285   ∈ wcel 1434  {cab 2069  Vcvv 2612

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614

This theorem is referenced by:  elpw  3412  elint  3668  opabid  4048  elrn2  4635  elimasn  4754  oprabid  5616  tfrlem3a  6007  tfrcllemsucaccv  6051  tfrcllembxssdm  6053  tfrcllemres  6059  addnqprlemrl  7019  addnqprlemru  7020  addnqprlemfl  7021  addnqprlemfu  7022  mulnqprlemrl  7035  mulnqprlemru  7036  mulnqprlemfl  7037  mulnqprlemfu  7038  ltnqpr  7055  ltnqpri  7056  archpr  7105  cauappcvgprlemladdfu  7116  cauappcvgprlemladdfl  7117  caucvgprlemladdfu  7139  caucvgprprlemopu  7161