Theorem hbun 2186

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 1008	. 2 |- ((y e. A \/ y e. B) -> A.x(y e. A \/ y e. B))
4		elun 2173	. 2 |- (y e. (A u. B) <-> (y e. A \/ y e. B))
5	4	albii 999	. 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 954   e. wcel 958   u. cun 2045

This theorem is referenced by:  trcl 4645  hbsum1 6983  hbsum 6984

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-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050

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