Theorem hbuni 2509

Description: Bound-variable hypothesis builder for union.

Hypothesis

Ref	Expression
hbuni.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbuni	|- (y e. U.A -> A.x y e. U.A)

Distinct variable groups:   y,A   x,y

Proof of Theorem hbuni

Step	Hyp	Ref	Expression
1		ax-17 971	. . . 4 |- (y e. z -> A.x y e. z)
2		hbuni.1	. . . . 5 |- (y e. A -> A.x y e. A)
3	1, 2	hbel 1566	. . . 4 |- (z e. A -> A.x z e. A)
4	1, 3	hban 1009	. . 3 |- ((y e. z /\ z e. A) -> A.x(y e. z /\ z e. A))
5	4	hbex 1006	. 2 |- (E.z(y e. z /\ z e. A) -> A.xE.z(y e. z /\ z e. A))
6		eluni 2506	. 2 |- (y e. U.A <-> E.z(y e. z /\ z e. A))
7	6	albii 999	. 2 |- (A.x y e. U.A <-> A.xE.z(y e. z /\ z e. A))
8	5, 6, 7	3imtr4 219	1 |- (y e. U.A -> A.x y e. U.A)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980  U.cuni 2503

This theorem is referenced by:  euuni 2881  reuuni2f 2883  reucl 2885  reuuni4 2887  reuuniss 2889  reuuniss2 2891  reuunixfr 2906  hbfv 3729  hbrdg 3936  trcl 4645  cardprc 4861  lble 6047  reuunineg 6066  hbsum1 6983  hbsum 6984  tgval3t 7625  minvecdist 8585  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-uni 2504

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