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Theorem cnveq 4557
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 4556 . . 3  |-  ( A 
C_  B  ->  `' A  C_  `' B )
2 cnvss 4556 . . 3  |-  ( B 
C_  A  ->  `' B  C_  `' A )
31, 2anim12i 331 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( `' A  C_  `' B  /\  `' B  C_  `' A ) )
4 eqss 3023 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3023 . 2  |-  ( `' A  =  `' B  <->  ( `' A  C_  `' B  /\  `' B  C_  `' A
) )
63, 4, 53imtr4i 199 1  |-  ( A  =  B  ->  `' A  =  `' B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    C_ wss 2982   `'ccnv 4390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-in 2988  df-ss 2995  df-br 3806  df-opab 3860  df-cnv 4399
This theorem is referenced by:  cnveqi  4558  cnveqd  4559  rneq  4609  cnveqb  4826  funcnvuni  5019  f1eq1  5138  f1o00  5212  foeqcnvco  5481  tposfn2  5935  ereq1  6200  infeq3  6522
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