Theorem hbun 2189

Description: Bound-variable hypothesis builder for the union of classes.

Hypotheses

Ref	Expression
hbun.1	|- (y e. A -> A.x y e. A)
hbun.2	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
hbun	|- (y e. (A u. B) -> A.x y e. (A u. B))

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem hbun

Step	Hyp	Ref	Expression
1		hbun.1	. . 3 |- (y e. A -> A.x y e. A)
2		hbun.2	. . 3 |- (y e. B -> A.x y e. B)
3	1, 2	hbor 1010	. 2 |- ((y e. A \/ y e. B) -> A.x(y e. A \/ y e. B))
4		elun 2176	. 2 |- (y e. (A u. B) <-> (y e. A \/ y e. B))
5	4	albii 1001	. 2 |- (A.x y e. (A u. B) <-> A.x(y e. A \/ y e. B))
6	3, 4, 5	3imtr4 219	1 |- (y e. (A u. B) -> A.x y e. (A u. B))

Syntax hints:   -> wi 3   \/ wo 222  A.wal 956   e. wcel 960   u. cun 2048

This theorem is referenced by:  trcl 4655  hbsum1 6983  hbsum 6984

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053

Copyright terms: Public domain