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Theorem bj-indeq 13232
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 13230 . 2 (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
2 df-bj-ind 13230 . . 3 (Ind 𝐵 ↔ (∅ ∈ 𝐵 ∧ ∀𝑥𝐵 suc 𝑥𝐵))
3 eleq2 2203 . . . . 5 (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
43bicomd 140 . . . 4 (𝐴 = 𝐵 → (∅ ∈ 𝐵 ↔ ∅ ∈ 𝐴))
5 eleq2 2203 . . . . . 6 (𝐴 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝑥𝐵))
65raleqbi1dv 2634 . . . . 5 (𝐴 = 𝐵 → (∀𝑥𝐴 suc 𝑥𝐴 ↔ ∀𝑥𝐵 suc 𝑥𝐵))
76bicomd 140 . . . 4 (𝐴 = 𝐵 → (∀𝑥𝐵 suc 𝑥𝐵 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
84, 7anbi12d 464 . . 3 (𝐴 = 𝐵 → ((∅ ∈ 𝐵 ∧ ∀𝑥𝐵 suc 𝑥𝐵) ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)))
92, 8syl5rbb 192 . 2 (𝐴 = 𝐵 → ((∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) ↔ Ind 𝐵))
101, 9syl5bb 191 1 (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2416  c0 3363  suc csuc 4287  Ind wind 13229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-bj-ind 13230
This theorem is referenced by:  bj-omind  13237  bj-omssind  13238  bj-ssom  13239  bj-om  13240  bj-2inf  13241
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