Theorem inex1g 2713

Description: Closed-form, generalized Separation Scheme.

Assertion

Ref	Expression
inex1g	|- (A e. C -> (A i^i B) e. V)

Proof of Theorem inex1g

Step	Hyp	Ref	Expression
1		ineq1 2206	. . 3 |- (x = A -> (x i^i B) = (A i^i B))
2	1	eleq1d 1537	. 2 |- (x = A -> ((x i^i B) e. V <-> (A i^i B) e. V))
3		visset 1809	. . 3 |- x e. V
4	3	inex1 2711	. 2 |- (x i^i B) e. V
5	2, 4	vtoclg 1843	1 |- (A e. C -> (A i^i B) e. V)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  Vcvv 1807   i^i cin 2042

This theorem is referenced by:  onin 2973  dmresexg 3374  resexg 3386  eltgt 7568  subtop 7596  infi1 10383  inpws1 10387  ficli 10404  filintf 10479  infi 10484

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047

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