Theorem bj-uniexg 15053

Description: uniexg 4453 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 3832	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2257	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2754	. . 3 |- x e. _V
4	3	bj-uniex 15052	. 2 |- U. x e. _V
5	2, 4	vtoclg 2811	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2159   _Vcvv 2751   U.cuni 3823

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-un 4447  ax-bd0 14948  ax-bdex 14954  ax-bdel 14956  ax-bdsb 14957  ax-bdsep 15019

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rex 2473  df-v 2753  df-uni 3824  df-bdc 14976

This theorem is referenced by: (None)