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  6232  offval3  6285  ssenen  7020  ressvalsets  13113  ressex  13114  ressbasd  13116  resseqnbasd  13122  ressinbasd  13123  ressressg  13124  qusin  13375  mgpress  13910  isunitd  14086  isrhm  14138  rhmfn  14152  rhmval  14153  2idlval  14482  2idlvalg  14483  eltg  14742  eltg3  14747  ntrval  14800  restco  14864  wlk1walkdom  16105