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Theorem cnveqb 5102
Description: Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
cnveqb ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))

Proof of Theorem cnveqb
StepHypRef Expression
1 cnveq 4819 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
2 dfrel2 5097 . . . 4 (Rel 𝐴𝐴 = 𝐴)
3 dfrel2 5097 . . . . . . 7 (Rel 𝐵𝐵 = 𝐵)
4 cnveq 4819 . . . . . . . . 9 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqeq2 2199 . . . . . . . . 9 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
64, 5imbitrrid 156 . . . . . . . 8 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
76eqcoms 2192 . . . . . . 7 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
83, 7sylbi 121 . . . . . 6 (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
9 eqeq1 2196 . . . . . . 7 (𝐴 = 𝐴 → (𝐴 = 𝐵𝐴 = 𝐵))
109imbi2d 230 . . . . . 6 (𝐴 = 𝐴 → ((𝐴 = 𝐵𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐴 = 𝐵)))
118, 10imbitrrid 156 . . . . 5 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1211eqcoms 2192 . . . 4 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
132, 12sylbi 121 . . 3 (Rel 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1413imp 124 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
151, 14impbid2 143 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  ccnv 4643  Rel wrel 4649
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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652
This theorem is referenced by:  cnveq0  5103
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