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Theorem cnveqb 5460
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 5174 . 2  |-  ( A  =  B  ->  `' A  =  `' B
)
2 dfrel2 5455 . . . 4  |-  ( Rel 
A  <->  `' `' A  =  A
)
3 dfrel2 5455 . . . . . . 7  |-  ( Rel 
B  <->  `' `' B  =  B
)
4 cnveq 5174 . . . . . . . . 9  |-  ( `' A  =  `' B  ->  `' `' A  =  `' `' B )
5 eqeq2 2482 . . . . . . . . 9  |-  ( B  =  `' `' B  ->  ( `' `' A  =  B  <->  `' `' A  =  `' `' B ) )
64, 5syl5ibr 221 . . . . . . . 8  |-  ( B  =  `' `' B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
76eqcoms 2479 . . . . . . 7  |-  ( `' `' B  =  B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
83, 7sylbi 195 . . . . . 6  |-  ( Rel 
B  ->  ( `' A  =  `' B  ->  `' `' A  =  B
) )
9 eqeq1 2471 . . . . . . 7  |-  ( A  =  `' `' A  ->  ( A  =  B  <->  `' `' A  =  B
) )
109imbi2d 316 . . . . . 6  |-  ( A  =  `' `' A  ->  ( ( `' A  =  `' B  ->  A  =  B )  <->  ( `' A  =  `' B  ->  `' `' A  =  B
) ) )
118, 10syl5ibr 221 . . . . 5  |-  ( A  =  `' `' A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
1211eqcoms 2479 . . . 4  |-  ( `' `' A  =  A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
132, 12sylbi 195 . . 3  |-  ( Rel 
A  ->  ( Rel  B  ->  ( `' A  =  `' B  ->  A  =  B ) ) )
1413imp 429 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  ( `' A  =  `' B  ->  A  =  B ) )
151, 14impbid2 204 1  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   `'ccnv 4998   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007
This theorem is referenced by:  cnveq0  5461  weisoeq2  6238
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