Theorem hbin 2220

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 groups:   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 1009	. 2 |- ((y e. A /\ y e. B) -> A.x(y e. A /\ y e. B))
4		elin 2207	. 2 |- (y e. (A i^i B) <-> (y e. A /\ y e. B))
5	4	albii 999	. 2 |- (A.x y e. (A i^i B) <-> A.x(y e. A /\ y e. B))
6	3, 4, 5	3imtr4 219	1 |- (y e. (A i^i B) -> A.x y e. (A i^i B))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958   i^i cin 2046

This theorem is referenced by:  hbres 3370  cp 4722

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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051

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