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Theorem cnveq 4427
Description: Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
Assertion
Ref Expression
cnveq (A = BA = B)

Proof of Theorem cnveq
StepHypRef Expression
1 cnvss 4426 . . 3 (ABAB)
2 cnvss 4426 . . 3 (BABA)
31, 2anim12i 321 . 2 ((AB BA) → (AB BA))
4 eqss 2931 . 2 (A = B ↔ (AB BA))
5 eqss 2931 . 2 (A = B ↔ (AB BA))
63, 4, 53imtr4i 190 1 (A = BA = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226  wss 2888  ccnv 4262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1622  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-in 2895  df-ss 2902  df-br 3731  df-opab 3785  df-cnv 4271
This theorem is referenced by:  cnveqi  4428  cnveqd  4429  rneq  4479  cnveqb  4694  funcnvuni  4885  f1eq1  5003  f1o00  5077  foeqcnvco  5346  tposfn2  5794  ereq1  6015
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