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Theorem cnveq 4436
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 4435 . . 3 (ABAB)
2 cnvss 4435 . . 3 (BABA)
31, 2anim12i 321 . 2 ((AB BA) → (AB BA))
4 eqss 2937 . 2 (A = B ↔ (AB BA))
5 eqss 2937 . 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 1228  wss 2894  ccnv 4271
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-in 2901  df-ss 2908  df-br 3739  df-opab 3793  df-cnv 4280
This theorem is referenced by:  cnveqi  4437  cnveqd  4438  rneq  4488  cnveqb  4703  funcnvuni  4894  f1eq1  5012  f1o00  5086  foeqcnvco  5355  tposfn2  5803  ereq1  6024
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