Theorem elab2 3323

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 3322	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  elpw  4114  elint  4416  opabid  4907  elrn2  5286  elimasn  5409  oprabid  6576  wfrlem3a  7304  tfrlem3a  7360  cardprclem  8688  iunfictbso  8820  aceq3lem  8826  dfac5lem4  8832  kmlem9  8863  domtriomlem  9147  ltexprlem3  9739  ltexprlem4  9740  reclem2pr  9749  reclem3pr  9750  supsrlem  9811  supaddc  10867  supadd  10868  supmul1  10869  supmullem1  10870  supmullem2  10871  supmul  10872  sqrlem6  13836  infcvgaux2i  14429  mertenslem1  14455  mertenslem2  14456  4sqlem12  15498  conjnmzb  17518  sylow3lem2  17866  mdetunilem9  20245  txuni2  21178  xkoopn  21202  met2ndci  22137  2sqlem8  24951  2sqlem11  24954  eulerpartlemt  29760  eulerpartlemr  29763  eulerpartlemn  29770  subfacp1lem3  30418  subfacp1lem5  30420  soseq  30995  nofulllem5  31105  finxpsuclem  32410  heiborlem1  32780  heiborlem6  32785  heiborlem8  32787  cllem0  36890