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 ∈ A → ∀x y ∈ A)
cleqh.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
cleqh	⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ 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 ↔ ∀y(y ∈ A ↔ y ∈ B))
2		ax-17 1616	. . . 4 ⊢ ((x ∈ A ↔ x ∈ B) → ∀y(x ∈ A ↔ x ∈ B))
3		dfbi2 609	. . . . 5 ⊢ ((y ∈ A ↔ y ∈ B) ↔ ((y ∈ A → y ∈ B) ∧ (y ∈ B → y ∈ A)))
4		cleqh.1	. . . . . . 7 ⊢ (y ∈ A → ∀x y ∈ A)
5		cleqh.2	. . . . . . 7 ⊢ (y ∈ B → ∀x y ∈ B)
6	4, 5	hbim 1817	. . . . . 6 ⊢ ((y ∈ A → y ∈ B) → ∀x(y ∈ A → y ∈ B))
7	5, 4	hbim 1817	. . . . . 6 ⊢ ((y ∈ B → y ∈ A) → ∀x(y ∈ B → y ∈ A))
8	6, 7	hban 1828	. . . . 5 ⊢ (((y ∈ A → y ∈ B) ∧ (y ∈ B → y ∈ A)) → ∀x((y ∈ A → y ∈ B) ∧ (y ∈ B → y ∈ A)))
9	3, 8	hbxfrbi 1568	. . . 4 ⊢ ((y ∈ A ↔ y ∈ B) → ∀x(y ∈ A ↔ y ∈ B))
10		eleq1 2413	. . . . . 6 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
11		eleq1 2413	. . . . . 6 ⊢ (x = y → (x ∈ B ↔ y ∈ B))
12	10, 11	bibi12d 312	. . . . 5 ⊢ (x = y → ((x ∈ A ↔ x ∈ B) ↔ (y ∈ A ↔ y ∈ B)))
13	12	biimpd 198	. . . 4 ⊢ (x = y → ((x ∈ A ↔ x ∈ B) → (y ∈ A ↔ y ∈ B)))
14	2, 9, 13	cbv3h 1983	. . 3 ⊢ (∀x(x ∈ A ↔ x ∈ B) → ∀y(y ∈ A ↔ y ∈ B))
15	12	equcoms 1681	. . . . 5 ⊢ (y = x → ((x ∈ A ↔ x ∈ B) ↔ (y ∈ A ↔ y ∈ B)))
16	15	biimprd 214	. . . 4 ⊢ (y = x → ((y ∈ A ↔ y ∈ B) → (x ∈ A ↔ x ∈ B)))
17	9, 2, 16	cbv3h 1983	. . 3 ⊢ (∀y(y ∈ A ↔ y ∈ B) → ∀x(x ∈ A ↔ x ∈ B))
18	14, 17	impbii 180	. 2 ⊢ (∀x(x ∈ A ↔ x ∈ B) ↔ ∀y(y ∈ A ↔ y ∈ B))
19	1, 18	bitr4i 243	1 ⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ 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