Theorem uniexd 4417

Description: Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
uniexd.1	|- ( ph -> A e. V )

Assertion

Ref	Expression
uniexd	|- ( ph -> U. A e. _V )

Proof of Theorem uniexd

Step	Hyp	Ref	Expression
1		uniexd.1	. 2 |- ( ph -> A e. V )
2		uniexg 4416	. 2 |- ( A e. V -> U. A e. _V )
3	1, 2	syl 14	1 |- ( ph -> U. A e. _V )

Syntax hints:   -> wi 4   e. wcel 2136   _Vcvv 2725   U.cuni 3788

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-sep 4099  ax-un 4410

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 2296  df-rex 2449  df-v 2727  df-uni 3789

This theorem is referenced by:  supex2g  6994