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Theorem bj-indeq 13271
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 13269 . 2  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
2 df-bj-ind 13269 . . 3  |-  (Ind  B  <->  (
(/)  e.  B  /\  A. x  e.  B  suc  x  e.  B )
)
3 eleq2 2203 . . . . 5  |-  ( A  =  B  ->  ( (/) 
e.  A  <->  (/)  e.  B
) )
43bicomd 140 . . . 4  |-  ( A  =  B  ->  ( (/) 
e.  B  <->  (/)  e.  A
) )
5 eleq2 2203 . . . . . 6  |-  ( A  =  B  ->  ( suc  x  e.  A  <->  suc  x  e.  B ) )
65raleqbi1dv 2634 . . . . 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 1331    e. wcel 1480   A.wral 2416   (/)c0 3363   suc csuc 4290  Ind wind 13268
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 13269
This theorem is referenced by:  bj-omind  13276  bj-omssind  13277  bj-ssom  13278  bj-om  13279  bj-2inf  13280
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