Theorem inex1g 4230

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 3403	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2300	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		vex 2806	. . 3 |- x e. _V
4	3	inex1 4228	. 2 |- ( x i^i B ) e. _V
5	2, 4	vtoclg 2865	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207

This theorem is referenced by:  onin  4489  dmresexg  5042  funimaexg  5421  offval  6252  offval3  6305  ssenen  7080  ressvalsets  13227  ressex  13228  ressbasd  13230  resseqnbasd  13236  ressinbasd  13237  ressressg  13238  qusin  13489  mgpress  14025  isunitd  14201  isrhm  14253  rhmfn  14267  rhmval  14268  2idlval  14598  2idlvalg  14599  eltg  14863  eltg3  14868  ntrval  14921  restco  14985  wlk1walkdom  16300