Theorem hbin 1647

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

Hypotheses

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

Assertion

Ref	Expression
hbin	|- (y e. (A i^i B) -> A.x y e. (A i^i B))

Distinct variable group(s):   y,A   y,B   x,y

Proof of Theorem hbin

Step	Hyp	Ref	Expression
1		hbin.1	. . 3 |- (y e. A -> A.x y e. A)
2		hbin.2	. . 3 |- (y e. B -> A.x y e. B)
3	1, 2	hban 704	. 2 |- ((y e. A /\ y e. B) -> A.x(y e. A /\ y e. B))
4		elin 1635	. 2 |- (y e. (A i^i B) <-> (y e. A /\ y e. B))
5	4	bial 695	. 2 |- (A.x y e. (A i^i B) <-> A.x(y e. A /\ y e. B))
6	3, 4, 5	3imtr4 192	1 |- (y e. (A i^i B) -> A.x y e. (A i^i B))

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   e. wcel 1092   i^i cin 1486

This theorem is referenced by:  hbres 2577  cp 3547

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-in 1491

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