Theorem inex1g 4059

Description: Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)

Assertion

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

Proof of Theorem inex1g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 3265	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2206	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		vex 2684	. . 3 |- x e. _V
4	3	inex1 4057	. 2 |- ( x i^i B ) e. _V
5	2, 4	vtoclg 2741	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2681   i^i cin 3065

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072

This theorem is referenced by:  onin  4303  dmresexg  4837  funimaexg  5202  offval  5982  offval3  6025  ssenen  6738  ressval2  12008  eltg  12210  eltg3  12215  ntrval  12268  restco  12332