Theorem bj-indeq 7299

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 7297	. 2 ⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A))
2		df-bj-ind 7297	. . 3 ⊢ (Ind B ↔ (∅ ∈ B ∧ ∀x ∈ B suc x ∈ B))
3		eleq2 2083	. . . . 5 ⊢ (A = B → (∅ ∈ A ↔ ∅ ∈ B))
4	3	bicomd 129	. . . 4 ⊢ (A = B → (∅ ∈ B ↔ ∅ ∈ A))
5		eleq2 2083	. . . . . 6 ⊢ (A = B → (suc x ∈ A ↔ suc x ∈ B))
6	5	raleqbi1dv 2491	. . . . 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 445	. . 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 1228   ∈ wcel 1374  ∀wral 2284  ∅c0 3201  suc csuc 4051  Ind wind 7296

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-bj-ind 7297

This theorem is referenced by:  bj-omind  7303  bj-omssind  7304  bj-ssom  7305  bj-om  7306  bj-2inf  7307