Theorem elab2 3652

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 3650	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2708  Vcvv 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804

This theorem is referenced by:  opabidw  5487  opabid  5488  oprabidw  7421  oprabid  7422  soseq  8141  tfrlem3a  8348  fsetfcdm  8836  cardprclem  9939  iunfictbso  10074  aceq3lem  10080  dfac5lem4  10086  dfac5lem4OLD  10088  kmlem9  10119  domtriomlem  10402  ltexprlem3  10998  ltexprlem4  10999  reclem2pr  11008  reclem3pr  11009  supsrlem  11071  supaddc  12157  supadd  12158  supmul1  12159  supmullem1  12160  supmullem2  12161  supmul  12162  01sqrexlem6  15220  infcvgaux2i  15831  mertenslem1  15857  mertenslem2  15858  4sqlem12  16934  conjnmzb  19192  sylow3lem2  19565  mdetunilem9  22514  txuni2  23459  xkoopn  23483  met2ndci  24417  2sqlem8  27344  2sqlem11  27347  madef  27771  eulerpartlemt  34369  eulerpartlemr  34372  eulerpartlemn  34379  subfacp1lem3  35176  subfacp1lem5  35178  rdgssun  37373  finxpsuclem  37392  heiborlem1  37812  heiborlem6  37817  heiborlem8  37819  cllem0  43562  brpermmodel  45000  fsetsnf  47056  fsetsnfo  47058  cfsetsnfsetf  47063  cfsetsnfsetf1  47064  cfsetsnfsetfo  47065