Theorem limeq 4080

Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
limeq	⊢ (A = B → (Lim A ↔ Lim B))

Proof of Theorem limeq

Step	Hyp	Ref	Expression
1		ordeq 4075	. . 3 ⊢ (A = B → (Ord A ↔ Ord B))
2		eleq2 2098	. . 3 ⊢ (A = B → (∅ ∈ A ↔ ∅ ∈ B))
3		id 19	. . . 4 ⊢ (A = B → A = B)
4		unieq 3580	. . . 4 ⊢ (A = B → ∪ A = ∪ B)
5	3, 4	eqeq12d 2051	. . 3 ⊢ (A = B → (A = ∪ A ↔ B = ∪ B))
6	1, 2, 5	3anbi123d 1206	. 2 ⊢ (A = B → ((Ord A ∧ ∅ ∈ A ∧ A = ∪ A) ↔ (Ord B ∧ ∅ ∈ B ∧ B = ∪ B)))
7		dflim2 4073	. 2 ⊢ (Lim A ↔ (Ord A ∧ ∅ ∈ A ∧ A = ∪ A))
8		dflim2 4073	. 2 ⊢ (Lim B ↔ (Ord B ∧ ∅ ∈ B ∧ B = ∪ B))
9	6, 7, 8	3bitr4g 212	1 ⊢ (A = B → (Lim A ↔ Lim B))

Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∅c0 3218  ∪ cuni 3571  Ord word 4065  Lim wlim 4067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-ilim 4072

This theorem is referenced by:  limuni2  4100