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Theorem cnveqb 4964
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 4683 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
2 dfrel2 4959 . . . 4 (Rel 𝐴𝐴 = 𝐴)
3 dfrel2 4959 . . . . . . 7 (Rel 𝐵𝐵 = 𝐵)
4 cnveq 4683 . . . . . . . . 9 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqeq2 2127 . . . . . . . . 9 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
64, 5syl5ibr 155 . . . . . . . 8 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
76eqcoms 2120 . . . . . . 7 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
83, 7sylbi 120 . . . . . 6 (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
9 eqeq1 2124 . . . . . . 7 (𝐴 = 𝐴 → (𝐴 = 𝐵𝐴 = 𝐵))
109imbi2d 229 . . . . . 6 (𝐴 = 𝐴 → ((𝐴 = 𝐵𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐴 = 𝐵)))
118, 10syl5ibr 155 . . . . 5 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1211eqcoms 2120 . . . 4 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
132, 12sylbi 120 . . 3 (Rel 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1413imp 123 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
151, 14impbid2 142 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  ccnv 4508  Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by:  cnveq0  4965
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