Theorem inex1g 4246

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 3415	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2301	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		vex 2816	. . 3 |- x e. _V
4	3	inex1 4244	. 2 |- ( x i^i B ) e. _V
5	2, 4	vtoclg 2875	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210

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 2214  ax-sep 4228

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217

This theorem is referenced by:  onin  4507  dmresexg  5061  funimaexg  5440  offval  6274  offval3  6327  ssenen  7105  hashfibclem  11206  ressvalsets  13277  ressex  13278  ressbasd  13280  resseqnbasd  13286  ressinbasd  13287  ressressg  13288  qusin  13539  mgpress  14075  isunitd  14251  isrhm  14303  rhmfn  14317  rhmval  14318  2idlval  14650  2idlvalg  14651  eltg  14917  eltg3  14922  ntrval  14975  restco  15039  wlk1walkdom  16354