Theorem inex1g 4118

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 3316	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵))
2	1	eleq1d 2235	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
3		vex 2729	. . 3 ⊢ 𝑥 ∈ V
4	3	inex1 4116	. 2 ⊢ (𝑥 ∩ 𝐵) ∈ V
5	2, 4	vtoclg 2786	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ∩ cin 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by:  onin  4364  dmresexg  4907  funimaexg  5272  offval  6057  offval3  6102  ssenen  6817  ressval2  12455  eltg  12692  eltg3  12697  ntrval  12750  restco  12814