Theorem inex1g 4251

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 3419	. . 3 |- ( x = A -> ( x i^i B ) = ( A i^i B ) )
2	1	eleq1d 2303	. 2 |- ( x = A -> ( ( x i^i B ) e. _V <-> ( A i^i B ) e. _V ) )
3		vex 2818	. . 3 |- x e. _V
4	3	inex1 4249	. 2 |- ( x i^i B ) e. _V
5	2, 4	vtoclg 2877	1 |- ( A e. V -> ( A i^i B ) e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   i^i cin 3213

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 2216  ax-sep 4233

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220

This theorem is referenced by:  onin  4512  dmresexg  5066  funimaexg  5445  offval  6283  offval3  6340  ssenen  7118  hashfibclem  11231  ressvalsets  13361  ressex  13362  ressbasd  13364  resseqnbasd  13370  ressinbasd  13371  ressressg  13372  qusin  13590  mgpress  14170  isunitd  14351  isrhm  14403  rhmfn  14417  rhmval  14418  2idlval  14776  2idlvalg  14777  eltg  15043  eltg3  15048  ntrval  15101  restco  15165  wlk1walkdom  16480