Theorem elab2 3354

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 3353	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  elpw  4164  elint  4481  opabid  4982  elrn2  5365  elimasn  5490  oprabid  6677  wfrlem3a  7417  tfrlem3a  7473  cardprclem  8805  iunfictbso  8937  aceq3lem  8943  dfac5lem4  8949  kmlem9  8980  domtriomlem  9264  ltexprlem3  9860  ltexprlem4  9861  reclem2pr  9870  reclem3pr  9871  supsrlem  9932  supaddc  10990  supadd  10991  supmul1  10992  supmullem1  10993  supmullem2  10994  supmul  10995  sqrlem6  13988  infcvgaux2i  14590  mertenslem1  14616  mertenslem2  14617  4sqlem12  15660  conjnmzb  17695  sylow3lem2  18043  mdetunilem9  20426  txuni2  21368  xkoopn  21392  met2ndci  22327  2sqlem8  25151  2sqlem11  25154  eulerpartlemt  30433  eulerpartlemr  30436  eulerpartlemn  30443  subfacp1lem3  31164  subfacp1lem5  31166  soseq  31751  finxpsuclem  33234  heiborlem1  33610  heiborlem6  33615  heiborlem8  33617  cllem0  37871