Theorem inex1g 3914

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 3160	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵))
2	1	eleq1d 2147	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
3		vex 2604	. . 3 ⊢ 𝑥 ∈ V
4	3	inex1 3912	. 2 ⊢ (𝑥 ∩ 𝐵) ∈ V
5	2, 4	vtoclg 2658	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∩ cin 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979

This theorem is referenced by:  onin  4141  dmresexg  4652  funimaexg  5003  offval  5739  offval3  5781