Theorem inex1g 4219

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 3398	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵))
2	1	eleq1d 2298	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
3		vex 2802	. . 3 ⊢ 𝑥 ∈ V
4	3	inex1 4217	. 2 ⊢ (𝑥 ∩ 𝐵) ∈ V
5	2, 4	vtoclg 2861	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∩ cin 3196

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 4201

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 2801  df-in 3203

This theorem is referenced by:  onin  4476  dmresexg  5027  funimaexg  5404  offval  6224  offval3  6277  ssenen  7008  ressvalsets  13092  ressex  13093  ressbasd  13095  resseqnbasd  13101  ressinbasd  13102  ressressg  13103  qusin  13354  mgpress  13889  isunitd  14064  isrhm  14116  rhmfn  14130  rhmval  14131  2idlval  14460  2idlvalg  14461  eltg  14720  eltg3  14725  ntrval  14778  restco  14842