Theorem inex1g 4220

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 3398	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2298	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		vex 2802	. . 3 |- x e. _V
4	3	inex1 4218	. 2 |- ( x i^i B ) e. _V
5	2, 4	vtoclg 2861	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4202

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203

This theorem is referenced by:  onin  4477  dmresexg  5028  funimaexg  5405  offval  6226  offval3  6279  ssenen  7012  ressvalsets  13097  ressex  13098  ressbasd  13100  resseqnbasd  13106  ressinbasd  13107  ressressg  13108  qusin  13359  mgpress  13894  isunitd  14070  isrhm  14122  rhmfn  14136  rhmval  14137  2idlval  14466  2idlvalg  14467  eltg  14726  eltg3  14731  ntrval  14784  restco  14848  wlk1walkdom  16070