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Theorem cnveq 4708
Description: Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
Assertion
Ref Expression
cnveq  |-  ( A  =  B  ->  `' A  =  `' B
)

Proof of Theorem cnveq
StepHypRef Expression
1 cnvss 4707 . . 3  |-  ( A 
C_  B  ->  `' A  C_  `' B )
2 cnvss 4707 . . 3  |-  ( B 
C_  A  ->  `' B  C_  `' A )
31, 2anim12i 336 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( `' A  C_  `' B  /\  `' B  C_  `' A ) )
4 eqss 3107 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3107 . 2  |-  ( `' A  =  `' B  <->  ( `' A  C_  `' B  /\  `' B  C_  `' A
) )
63, 4, 53imtr4i 200 1  |-  ( A  =  B  ->  `' A  =  `' B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    C_ wss 3066   `'ccnv 4533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-br 3925  df-opab 3985  df-cnv 4542
This theorem is referenced by:  cnveqi  4709  cnveqd  4710  rneq  4761  cnveqb  4989  funcnvuni  5187  f1eq1  5318  f1o00  5395  foeqcnvco  5684  tposfn2  6156  ereq1  6429  infeq3  6895  iscn  12351  ishmeo  12458
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