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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.
StepHypRef 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))
43bicomd 129 . . . 4 (A = B → (∅ B ↔ ∅ A))
5 eleq2 2083 . . . . . 6 (A = B → (suc x A ↔ suc x B))
65raleqbi1dv 2491 . . . . 5 (A = B → (x A suc x Ax B suc x B))
76bicomd 129 . . . 4 (A = B → (x B suc x Bx A suc x A))
84, 7anbi12d 445 . . 3 (A = B → ((∅ B x B suc x B) ↔ (∅ A x A suc x A)))
92, 8syl5rbb 182 . 2 (A = B → ((∅ A x A suc x A) ↔ Ind B))
101, 9syl5bb 181 1 (A = B → (Ind A ↔ Ind B))
Colors of variables: wff set class
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
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