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Theorem cnveq 4819
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 4818 . . 3 (𝐴𝐵𝐴𝐵)
2 cnvss 4818 . . 3 (𝐵𝐴𝐵𝐴)
31, 2anim12i 338 . 2 ((𝐴𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))
4 eqss 3185 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3185 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
63, 4, 53imtr4i 201 1 (𝐴 = 𝐵𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wss 3144  ccnv 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157  df-br 4019  df-opab 4080  df-cnv 4652
This theorem is referenced by:  cnveqi  4820  cnveqd  4821  rneq  4872  cnveqb  5102  funcnvuni  5304  f1eq1  5435  f1o00  5515  foeqcnvco  5811  tposfn2  6290  ereq1  6565  infeq3  7043  1arith  12398  isrim0  13508  psrbag  13944  iscn  14149  ishmeo  14256
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