Theorem hbdif 2164

Description: Bound-variable hypothesis builder for class difference.

Hypotheses

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

Assertion

Ref	Expression
hbdif	|- (y e. (A \ B) -> A.x y e. (A \ B))

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

Proof of Theorem hbdif

Step	Hyp	Ref	Expression
1		hbdif.1	. . 3 |- (y e. A -> A.x y e. A)
2		hbdif.2	. . . 4 |- (y e. B -> A.x y e. B)
3	2	hbn 1006	. . 3 |- (-. y e. B -> A.x -. y e. B)
4	1, 3	hban 1011	. 2 |- ((y e. A /\ -. y e. B) -> A.x(y e. A /\ -. y e. B))
5		eldif 2060	. 2 |- (y e. (A \ B) <-> (y e. A /\ -. y e. B))
6	5	albii 1001	. 2 |- (A.x y e. (A \ B) <-> A.x(y e. A /\ -. y e. B))
7	4, 5, 6	3imtr4 219	1 |- (y e. (A \ B) -> A.x y e. (A \ B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 956   e. wcel 960   \ cdif 2047

This theorem is referenced by:  unblem2 4552  unblem3 4553

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-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-dif 2052

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