Theorem cleqh 2178

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2242. (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 2075	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
2		ax-17 1459	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		dfbi2 380	. . . . 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 1477	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑥(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵))
7	5, 4	hbim 1477	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴) → ∀𝑥(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴))
8	6, 7	hban 1479	. . . . 5 ⊢ (((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)) → ∀𝑥((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
9	3, 8	hbxfrbi 1401	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → ∀𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
10		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
11		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
12	10, 11	bibi12d 233	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)))
13	12	biimpd 142	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)))
14	2, 9, 13	cbv3h 1671	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
15	12	equcoms 1634	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)))
16	15	biimprd 156	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
17	9, 2, 16	cbv3h 1671	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
18	14, 17	impbii 124	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
19	1, 18	bitr4i 185	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284   ∈ wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077

This theorem is referenced by:  abeq2  2187