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Theorem bj-indeq 13054
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 13052 . 2  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
2 df-bj-ind 13052 . . 3  |-  (Ind  B  <->  (
(/)  e.  B  /\  A. x  e.  B  suc  x  e.  B )
)
3 eleq2 2181 . . . . 5  |-  ( A  =  B  ->  ( (/) 
e.  A  <->  (/)  e.  B
) )
43bicomd 140 . . . 4  |-  ( A  =  B  ->  ( (/) 
e.  B  <->  (/)  e.  A
) )
5 eleq2 2181 . . . . . 6  |-  ( A  =  B  ->  ( suc  x  e.  A  <->  suc  x  e.  B ) )
65raleqbi1dv 2611 . . . . 5  |-  ( A  =  B  ->  ( A. x  e.  A  suc  x  e.  A  <->  A. x  e.  B  suc  x  e.  B ) )
76bicomd 140 . . . 4  |-  ( A  =  B  ->  ( A. x  e.  B  suc  x  e.  B  <->  A. x  e.  A  suc  x  e.  A ) )
84, 7anbi12d 464 . . 3  |-  ( A  =  B  ->  (
( (/)  e.  B  /\  A. x  e.  B  suc  x  e.  B )  <->  (
(/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
) )
92, 8syl5rbb 192 . 2  |-  ( A  =  B  ->  (
( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  <-> 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 1316    e. wcel 1465   A.wral 2393   (/)c0 3333   suc csuc 4257  Ind wind 13051
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-bj-ind 13052
This theorem is referenced by:  bj-omind  13059  bj-omssind  13060  bj-ssom  13061  bj-om  13062  bj-2inf  13063
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