Theorem cleqh 2450

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)

Hypotheses

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

Assertion

Ref	Expression
cleqh	|- (A = B <-> A.x(x e. A <-> x e. B))

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

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem cleqh

Step	Hyp	Ref	Expression
1		dfcleq 2347	. 2 |- (A = B <-> A.y(y e. A <-> y e. B))
2		ax-17 1616	. . . 4 |- ((x e. A <-> x e. B) -> A.y(x e. A <-> x e. B))
3		dfbi2 609	. . . . 5 |- ((y e. A <-> y e. B) <-> ((y e. A -> y e. B) /\ (y e. B -> y e. A)))
4		cleqh.1	. . . . . . 7 |- (y e. A -> A.x y e. A)
5		cleqh.2	. . . . . . 7 |- (y e. B -> A.x y e. B)
6	4, 5	hbim 1817	. . . . . 6 |- ((y e. A -> y e. B) -> A.x(y e. A -> y e. B))
7	5, 4	hbim 1817	. . . . . 6 |- ((y e. B -> y e. A) -> A.x(y e. B -> y e. A))
8	6, 7	hban 1828	. . . . 5 |- (((y e. A -> y e. B) /\ (y e. B -> y e. A)) -> A.x((y e. A -> y e. B) /\ (y e. B -> y e. A)))
9	3, 8	hbxfrbi 1568	. . . 4 |- ((y e. A <-> y e. B) -> A.x(y e. A <-> y e. B))
10		eleq1 2413	. . . . . 6 |- (x = y -> (x e. A <-> y e. A))
11		eleq1 2413	. . . . . 6 |- (x = y -> (x e. B <-> y e. B))
12	10, 11	bibi12d 312	. . . . 5 |- (x = y -> ((x e. A <-> x e. B) <-> (y e. A <-> y e. B)))
13	12	biimpd 198	. . . 4 |- (x = y -> ((x e. A <-> x e. B) -> (y e. A <-> y e. B)))
14	2, 9, 13	cbv3h 1983	. . 3 |- (A.x(x e. A <-> x e. B) -> A.y(y e. A <-> y e. B))
15	12	equcoms 1681	. . . . 5 |- (y = x -> ((x e. A <-> x e. B) <-> (y e. A <-> y e. B)))
16	15	biimprd 214	. . . 4 |- (y = x -> ((y e. A <-> y e. B) -> (x e. A <-> x e. B)))
17	9, 2, 16	cbv3h 1983	. . 3 |- (A.y(y e. A <-> y e. B) -> A.x(x e. A <-> x e. B))
18	14, 17	impbii 180	. 2 |- (A.x(x e. A <-> x e. B) <-> A.y(y e. A <-> y e. B))
19	1, 18	bitr4i 243	1 |- (A = B <-> A.x(x e. A <-> x e. B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349

This theorem is referenced by:  abeq2  2459