Theorem bj-indeq 9388

Description: Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-indeq	⊢ (A = B → (Ind A ↔ Ind B))

Proof of Theorem bj-indeq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-ind 9386	. 2 ⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A))
2		df-bj-ind 9386	. . 3 ⊢ (Ind B ↔ (∅ ∈ B ∧ ∀x ∈ B suc x ∈ B))
3		eleq2 2098	. . . . 5 ⊢ (A = B → (∅ ∈ A ↔ ∅ ∈ B))
4	3	bicomd 129	. . . 4 ⊢ (A = B → (∅ ∈ B ↔ ∅ ∈ A))
5		eleq2 2098	. . . . . 6 ⊢ (A = B → (suc x ∈ A ↔ suc x ∈ B))
6	5	raleqbi1dv 2507	. . . . 5 ⊢ (A = B → (∀x ∈ A suc x ∈ A ↔ ∀x ∈ B suc x ∈ B))
7	6	bicomd 129	. . . 4 ⊢ (A = B → (∀x ∈ B suc x ∈ B ↔ ∀x ∈ A suc x ∈ A))
8	4, 7	anbi12d 442	. . 3 ⊢ (A = B → ((∅ ∈ B ∧ ∀x ∈ B suc x ∈ B) ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)))
9	2, 8	syl5rbb 182	. 2 ⊢ (A = B → ((∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) ↔ Ind B))
10	1, 9	syl5bb 181	1 ⊢ (A = B → (Ind A ↔ Ind B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∅c0 3218  suc csuc 4068  Ind wind 9385

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-bndl 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-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-bj-ind 9386

This theorem is referenced by:  bj-omind  9393  bj-omssind  9394  bj-ssom  9395  bj-om  9396  bj-2inf  9397  peano5setOLD  9400