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Theorem cnveqb 5278
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 4996 . 2  |-  ( A  =  B  ->  `' A  =  `' B
)
2 dfrel2 5273 . . . 4  |-  ( Rel 
A  <->  `' `' A  =  A
)
3 dfrel2 5273 . . . . . . 7  |-  ( Rel 
B  <->  `' `' B  =  B
)
4 cnveq 4996 . . . . . . . . 9  |-  ( `' A  =  `' B  ->  `' `' A  =  `' `' B )
5 eqeq2 2417 . . . . . . . . 9  |-  ( B  =  `' `' B  ->  ( `' `' A  =  B  <->  `' `' A  =  `' `' B ) )
64, 5syl5ibr 221 . . . . . . . 8  |-  ( B  =  `' `' B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
76eqcoms 2414 . . . . . . 7  |-  ( `' `' B  =  B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
83, 7sylbi 195 . . . . . 6  |-  ( Rel 
B  ->  ( `' A  =  `' B  ->  `' `' A  =  B
) )
9 eqeq1 2406 . . . . . . 7  |-  ( A  =  `' `' A  ->  ( A  =  B  <->  `' `' A  =  B
) )
109imbi2d 314 . . . . . 6  |-  ( A  =  `' `' A  ->  ( ( `' A  =  `' B  ->  A  =  B )  <->  ( `' A  =  `' B  ->  `' `' A  =  B
) ) )
118, 10syl5ibr 221 . . . . 5  |-  ( A  =  `' `' A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
1211eqcoms 2414 . . . 4  |-  ( `' `' A  =  A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
132, 12sylbi 195 . . 3  |-  ( Rel 
A  ->  ( Rel  B  ->  ( `' A  =  `' B  ->  A  =  B ) ) )
1413imp 427 . 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 367    = wceq 1405   `'ccnv 4821   Rel wrel 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-cnv 4830
This theorem is referenced by:  cnveq0  5279  weisoeq2  6234  relexpaddg  13033  relexpaddss  35677
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