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Theorem cnveq 4761
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 4760 . . 3 (𝐴𝐵𝐴𝐵)
2 cnvss 4760 . . 3 (𝐵𝐴𝐵𝐴)
31, 2anim12i 336 . 2 ((𝐴𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))
4 eqss 3143 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3143 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
63, 4, 53imtr4i 200 1 (𝐴 = 𝐵𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wss 3102  ccnv 4586
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-br 3967  df-opab 4027  df-cnv 4595
This theorem is referenced by:  cnveqi  4762  cnveqd  4763  rneq  4814  cnveqb  5042  funcnvuni  5240  f1eq1  5371  f1o00  5450  foeqcnvco  5741  tposfn2  6214  ereq1  6488  infeq3  6960  iscn  12639  ishmeo  12746
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