Theorem uniexd 4475

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 4474	. 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 2167   _Vcvv 2763   U.cuni 3839

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-uni 3840

This theorem is referenced by:  supex2g  7099  ptex  12935  zrhval  14173  zrhvalg  14174  zrhex  14177