Theorem hbuni 1925

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 group(s):   y,A   x,y

Proof of Theorem hbuni

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 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 1172	. . . 4 |- (z e. A -> A.x z e. A)
4	1, 3	hban 704	. . 3 |- ((y e. z /\ z e. A) -> A.x(y e. z /\ z e. A))
5	4	hbex 701	. 2 |- (E.z(y e. z /\ z e. A) -> A.xE.z(y e. z /\ z e. A))
6		eluni 1922	. 2 |- (y e. U.A <-> E.z(y e. z /\ z e. A))
7	6	bial 695	. 2 |- (A.x y e. U.A <-> A.xE.z(y e. z /\ z e. A))
8	5, 6, 7	3imtr4 192	1 |- (y e. U.A -> A.x y e. U.A)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803   e. wcel 1092  U.cuni 1919

This theorem is referenced by:  euuni 1954  reuuni2 1956  reucl 1957  reuuni4 1959  hbfv 2837  hbrdg 2974  trcl 3489  cardprc 3667

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-uni 1920

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