Theorem inex1g 4225

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 3401	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝐵) = (𝐴 ∩ 𝐵))
2	1	eleq1d 2300	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝐵) ∈ V ↔ (𝐴 ∩ 𝐵) ∈ V))
3		vex 2805	. . 3 ⊢ 𝑥 ∈ V
4	3	inex1 4223	. 2 ⊢ (𝑥 ∩ 𝐵) ∈ V
5	2, 4	vtoclg 2864	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∩ cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  onin  4483  dmresexg  5036  funimaexg  5414  offval  6242  offval3  6295  ssenen  7036  ressvalsets  13146  ressex  13147  ressbasd  13149  resseqnbasd  13155  ressinbasd  13156  ressressg  13157  qusin  13408  mgpress  13943  isunitd  14119  isrhm  14171  rhmfn  14185  rhmval  14186  2idlval  14515  2idlvalg  14516  eltg  14775  eltg3  14780  ntrval  14833  restco  14897  wlk1walkdom  16209