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Theorem cnveq 4610
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 4609 . . 3 (𝐴𝐵𝐴𝐵)
2 cnvss 4609 . . 3 (𝐵𝐴𝐵𝐴)
31, 2anim12i 331 . 2 ((𝐴𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))
4 eqss 3040 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3040 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
63, 4, 53imtr4i 199 1 (𝐴 = 𝐵𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wss 2999  ccnv 4437
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3005  df-ss 3012  df-br 3846  df-opab 3900  df-cnv 4446
This theorem is referenced by:  cnveqi  4611  cnveqd  4612  rneq  4662  cnveqb  4886  funcnvuni  5083  f1eq1  5211  f1o00  5288  foeqcnvco  5569  tposfn2  6031  ereq1  6297  infeq3  6708
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