Theorem inex1g 4223

Description: Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)

Assertion

Ref	Expression
inex1g	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Proof of Theorem inex1g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 3399	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵))
2	1	eleq1d 2298	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
3		vex 2803	. . 3 ⊢ 𝑥 ∈ V
4	3	inex1 4221	. 2 ⊢ (𝑥 ∩ 𝐵) ∈ V
5	2, 4	vtoclg 2862	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∩ cin 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4205

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-in 3204

This theorem is referenced by:  onin  4481  dmresexg  5034  funimaexg  5411  offval  6238  offval3  6291  ssenen  7032  ressvalsets  13137  ressex  13138  ressbasd  13140  resseqnbasd  13146  ressinbasd  13147  ressressg  13148  qusin  13399  mgpress  13934  isunitd  14110  isrhm  14162  rhmfn  14176  rhmval  14177  2idlval  14506  2idlvalg  14507  eltg  14766  eltg3  14771  ntrval  14824  restco  14888  wlk1walkdom  16156