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Theorem cnveqb 5625
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 5328 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
2 dfrel2 5618 . . . 4 (Rel 𝐴𝐴 = 𝐴)
3 dfrel2 5618 . . . . . . 7 (Rel 𝐵𝐵 = 𝐵)
4 cnveq 5328 . . . . . . . . 9 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqeq2 2662 . . . . . . . . 9 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
64, 5syl5ibr 236 . . . . . . . 8 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
76eqcoms 2659 . . . . . . 7 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
83, 7sylbi 207 . . . . . 6 (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
9 eqeq1 2655 . . . . . . 7 (𝐴 = 𝐴 → (𝐴 = 𝐵𝐴 = 𝐵))
109imbi2d 329 . . . . . 6 (𝐴 = 𝐴 → ((𝐴 = 𝐵𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐴 = 𝐵)))
118, 10syl5ibr 236 . . . . 5 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1211eqcoms 2659 . . . 4 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
132, 12sylbi 207 . . 3 (Rel 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1413imp 444 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
151, 14impbid2 216 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  ccnv 5142  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151
This theorem is referenced by:  cnveq0  5626  weisoeq2  6646  relexpaddg  13837  relexpaddss  38327
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