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Theorem cnveq 4783
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 4782 . . 3  |-  ( A 
C_  B  ->  `' A  C_  `' B )
2 cnvss 4782 . . 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 3162 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3162 . 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 1348    C_ wss 3121   `'ccnv 4608
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3988  df-opab 4049  df-cnv 4617
This theorem is referenced by:  cnveqi  4784  cnveqd  4785  rneq  4836  cnveqb  5064  funcnvuni  5265  f1eq1  5396  f1o00  5475  foeqcnvco  5767  tposfn2  6243  ereq1  6518  infeq3  6990  1arith  12312  iscn  12956  ishmeo  13063
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