ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnveq Unicode version

Theorem cnveq 4683
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 4682 . . 3  |-  ( A 
C_  B  ->  `' A  C_  `' B )
2 cnvss 4682 . . 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 3082 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 eqss 3082 . 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 1316    C_ wss 3041   `'ccnv 4508
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-in 3047  df-ss 3054  df-br 3900  df-opab 3960  df-cnv 4517
This theorem is referenced by:  cnveqi  4684  cnveqd  4685  rneq  4736  cnveqb  4964  funcnvuni  5162  f1eq1  5293  f1o00  5370  foeqcnvco  5659  tposfn2  6131  ereq1  6404  infeq3  6870  iscn  12293  ishmeo  12400
  Copyright terms: Public domain W3C validator