Theorem bj-uniexg 15415

Description: uniexg 4470 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 3844	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2262	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2763	. . 3 |- x e. _V
4	3	bj-uniex 15414	. 2 |- U. x e. _V
5	2, 4	vtoclg 2820	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   U.cuni 3835

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-un 4464  ax-bd0 15310  ax-bdex 15316  ax-bdel 15318  ax-bdsb 15319  ax-bdsep 15381

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-uni 3836  df-bdc 15338

This theorem is referenced by: (None)