Theorem inex1g 1699

Description: Closed-form, generalized Separation Scheme.

Assertion

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

Proof of Theorem inex1g

Step	Hyp	Ref	Expression
1		ineq1 1638	. . 3 ⊢ (x = A → (x ∩ B) = (A ∩ B))
2	1	eleq1d 1155	. 2 ⊢ (x = A → ((x ∩ B) ∈ V ↔ (A ∩ B) ∈ V))
3		visset 1350	. . 3 ⊢ x ∈ V
4	3	inex1 1697	. 2 ⊢ (x ∩ B) ∈ V
5	2, 4	vtoclg 1383	1 ⊢ (A ∈ C → (A ∩ B) ∈ V)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486

This theorem is referenced by:  onin 2229  dmresexg 2586  resexg 2597

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491

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