Theorem inex1g 4271

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 4269	. 2 |- ( x i^i B ) e. _V
5	2, 4	vtoclg 2883	1 |- ( A e. V -> ( A i^i B ) e. _V )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219

This proof depends on 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 4251

This proof 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 used by:  onin  4533  dmresexg  5088  funimaexg  5468  ofovex  6311  offval  6314  offval3  6371  ssenen  7156  hashfibclem  11304  ressvalsets  13476  ressex  13477  ressbasd  13480  resseqnbasd  13486  ressinbasd  13487  ressressg  13488  qusin  13706  mgpress  14337  isunitd  14522  isrhm  14574  rhmfn  14588  rhmval  14589  2idlval  14948  2idlvalg  14949  eltg  15278  eltg3  15283  ntrval  15336  restco  15400  wlk1walkdom  16805