Theorem inex1g 4264

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 3425	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2307	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		vex 2824	. . 3 |- x e. _V
4	3	inex1 4262	. 2 |- ( x i^i B ) e. _V
5	2, 4	vtoclg 2883	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is referenced by:  onin  4526  dmresexg  5081  funimaexg  5460  offval  6300  offval3  6357  ssenen  7142  hashfibclem  11260  ressvalsets  13395  ressex  13396  ressbasd  13398  resseqnbasd  13404  ressinbasd  13405  ressressg  13406  qusin  13624  mgpress  14205  isunitd  14386  isrhm  14438  rhmfn  14452  rhmval  14453  2idlval  14811  2idlvalg  14812  eltg  15076  eltg3  15081  ntrval  15134  restco  15198  wlk1walkdom  16514