Theorem bj-uniexg 13800

Description: uniexg 4417 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-uniexg	|- ( A e. V -> U. A e. _V )

Proof of Theorem bj-uniexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 3798	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2235	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2729	. . 3 |- x e. _V
4	3	bj-uniex 13799	. 2 |- U. x e. _V
5	2, 4	vtoclg 2786	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   U.cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-un 4411  ax-bd0 13695  ax-bdex 13701  ax-bdel 13703  ax-bdsb 13704  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-uni 3790  df-bdc 13723

This theorem is referenced by: (None)