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Theorem cnveqb 5129
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 4855 . 2  |-  ( A  =  B  ->  `' A  =  `' B
)
2 dfrel2 5124 . . . 4  |-  ( Rel 
A  <->  `' `' A  =  A
)
3 dfrel2 5124 . . . . . . 7  |-  ( Rel 
B  <->  `' `' B  =  B
)
4 cnveq 4855 . . . . . . . . 9  |-  ( `' A  =  `' B  ->  `' `' A  =  `' `' B )
5 eqeq2 2292 . . . . . . . . 9  |-  ( B  =  `' `' B  ->  ( `' `' A  =  B  <->  `' `' A  =  `' `' B ) )
64, 5syl5ibr 212 . . . . . . . 8  |-  ( B  =  `' `' B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
76eqcoms 2286 . . . . . . 7  |-  ( `' `' B  =  B  ->  ( `' A  =  `' B  ->  `' `' A  =  B )
)
83, 7sylbi 187 . . . . . 6  |-  ( Rel 
B  ->  ( `' A  =  `' B  ->  `' `' A  =  B
) )
9 eqeq1 2289 . . . . . . 7  |-  ( A  =  `' `' A  ->  ( A  =  B  <->  `' `' A  =  B
) )
109imbi2d 307 . . . . . 6  |-  ( A  =  `' `' A  ->  ( ( `' A  =  `' B  ->  A  =  B )  <->  ( `' A  =  `' B  ->  `' `' A  =  B
) ) )
118, 10syl5ibr 212 . . . . 5  |-  ( A  =  `' `' A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
1211eqcoms 2286 . . . 4  |-  ( `' `' A  =  A  ->  ( Rel  B  -> 
( `' A  =  `' B  ->  A  =  B ) ) )
132, 12sylbi 187 . . 3  |-  ( Rel 
A  ->  ( Rel  B  ->  ( `' A  =  `' B  ->  A  =  B ) ) )
1413imp 418 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  ( `' A  =  `' B  ->  A  =  B ) )
151, 14impbid2 195 1  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   `'ccnv 4688   Rel wrel 4694
This theorem is referenced by:  cnveq0  5130  weisoeq2  5854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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