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Theorem cnveqb 5267
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 4987 . 2  |-  ( A  =  B  ->  `' A  =  `' B
)
2 dfrel2 5262 . . . 4  |-  ( Rel 
A  <->  `' `' A  =  A
)
3 dfrel2 5262 . . . . . . 7  |-  ( Rel 
B  <->  `' `' B  =  B
)
4 cnveq 4987 . . . . . . . . 9  |-  ( `' A  =  `' B  ->  `' `' A  =  `' `' B )
5 eqeq2 2397 . . . . . . . . 9  |-  ( B  =  `' `' B  ->  ( `' `' A  =  B  <->  `' `' A  =  `' `' B ) )
64, 5syl5ibr 213 . . . . . . . 8  |-  ( B  =  `' `' B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
76eqcoms 2391 . . . . . . 7  |-  ( `' `' B  =  B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
83, 7sylbi 188 . . . . . 6  |-  ( Rel 
B  ->  ( `' A  =  `' B  ->  `' `' A  =  B
) )
9 eqeq1 2394 . . . . . . 7  |-  ( A  =  `' `' A  ->  ( A  =  B  <->  `' `' A  =  B
) )
109imbi2d 308 . . . . . 6  |-  ( A  =  `' `' A  ->  ( ( `' A  =  `' B  ->  A  =  B )  <->  ( `' A  =  `' B  ->  `' `' A  =  B
) ) )
118, 10syl5ibr 213 . . . . 5  |-  ( A  =  `' `' A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
1211eqcoms 2391 . . . 4  |-  ( `' `' A  =  A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
132, 12sylbi 188 . . 3  |-  ( Rel 
A  ->  ( Rel  B  ->  ( `' A  =  `' B  ->  A  =  B ) ) )
1413imp 419 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  ( `' A  =  `' B  ->  A  =  B ) )
151, 14impbid2 196 1  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   `'ccnv 4818   Rel wrel 4824
This theorem is referenced by:  cnveq0  5268  weisoeq2  6017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827
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