Theorem hbdif 1590

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 group(s):   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	hbne 699	. . 3 |- (-. y e. B -> A.x -. y e. B)
4	1, 3	hban 704	. 2 |- ((y e. A /\ -. y e. B) -> A.x(y e. A /\ -. y e. B))
5		eldif 1496	. 2 |- (y e. (A \ B) <-> (y e. A /\ -. y e. B))
6	5	bial 695	. 2 |- (A.x y e. (A \ B) <-> A.x(y e. A /\ -. y e. B))
7	4, 5, 6	3imtr4 192	1 |- (y e. (A \ B) -> A.x y e. (A \ B))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672   e. wcel 1092   \ cdif 1484

This theorem is referenced by:  unblem2 3432  unblem3 3433

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-dif 1489

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