Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-indeq Structured version   GIF version

Theorem bj-indeq 9364
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 9362 . 2 (Ind A ↔ (∅ A x A suc x A))
2 df-bj-ind 9362 . . 3 (Ind B ↔ (∅ B x B suc x B))
3 eleq2 2098 . . . . 5 (A = B → (∅ A ↔ ∅ B))
43bicomd 129 . . . 4 (A = B → (∅ B ↔ ∅ A))
5 eleq2 2098 . . . . . 6 (A = B → (suc x A ↔ suc x B))
65raleqbi1dv 2507 . . . . 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 442 . . 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 1242   wcel 1390  wral 2300  c0 3218  suc csuc 4068  Ind wind 9361
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-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-bj-ind 9362
This theorem is referenced by:  bj-omind  9368  bj-omssind  9369  bj-ssom  9370  bj-om  9371  bj-2inf  9372
  Copyright terms: Public domain W3C validator