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Theorem bj-indeq 13886
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 eleq2 2234 . . 3  |-  ( A  =  B  ->  ( (/) 
e.  A  <->  (/)  e.  B
) )
2 eleq2 2234 . . . 4  |-  ( A  =  B  ->  ( suc  x  e.  A  <->  suc  x  e.  B ) )
32raleqbi1dv 2673 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  suc  x  e.  A  <->  A. x  e.  B  suc  x  e.  B ) )
41, 3anbi12d 470 . 2  |-  ( A  =  B  ->  (
( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  <->  (
(/)  e.  B  /\  A. x  e.  B  suc  x  e.  B )
) )
5 df-bj-ind 13884 . 2  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
6 df-bj-ind 13884 . 2  |-  (Ind  B  <->  (
(/)  e.  B  /\  A. x  e.  B  suc  x  e.  B )
)
74, 5, 63bitr4g 222 1  |-  ( A  =  B  ->  (Ind  A 
<-> Ind 
B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448   (/)c0 3414   suc csuc 4348  Ind wind 13883
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-bj-ind 13884
This theorem is referenced by:  bj-omind  13891  bj-omssind  13892  bj-ssom  13893  bj-om  13894  bj-2inf  13895
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