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Theorem cnveqb 4827
Description: Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
cnveqb  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )

Proof of Theorem cnveqb
StepHypRef Expression
1 cnveq 4558 . 2  |-  ( A  =  B  ->  `' A  =  `' B
)
2 dfrel2 4822 . . . 4  |-  ( Rel 
A  <->  `' `' A  =  A
)
3 dfrel2 4822 . . . . . . 7  |-  ( Rel 
B  <->  `' `' B  =  B
)
4 cnveq 4558 . . . . . . . . 9  |-  ( `' A  =  `' B  ->  `' `' A  =  `' `' B )
5 eqeq2 2092 . . . . . . . . 9  |-  ( B  =  `' `' B  ->  ( `' `' A  =  B  <->  `' `' A  =  `' `' B ) )
64, 5syl5ibr 154 . . . . . . . 8  |-  ( B  =  `' `' B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
76eqcoms 2086 . . . . . . 7  |-  ( `' `' B  =  B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
83, 7sylbi 119 . . . . . 6  |-  ( Rel 
B  ->  ( `' A  =  `' B  ->  `' `' A  =  B
) )
9 eqeq1 2089 . . . . . . 7  |-  ( A  =  `' `' A  ->  ( A  =  B  <->  `' `' A  =  B
) )
109imbi2d 228 . . . . . 6  |-  ( A  =  `' `' A  ->  ( ( `' A  =  `' B  ->  A  =  B )  <->  ( `' A  =  `' B  ->  `' `' A  =  B
) ) )
118, 10syl5ibr 154 . . . . 5  |-  ( A  =  `' `' A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
1211eqcoms 2086 . . . 4  |-  ( `' `' A  =  A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
132, 12sylbi 119 . . 3  |-  ( Rel 
A  ->  ( Rel  B  ->  ( `' A  =  `' B  ->  A  =  B ) ) )
1413imp 122 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  ( `' A  =  `' B  ->  A  =  B ) )
151, 14impbid2 141 1  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   `'ccnv 4391   Rel wrel 4397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-rel 4399  df-cnv 4400
This theorem is referenced by:  cnveq0  4828
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