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Theorem cnveqb 5731
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 5434 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
2 dfrel2 5724 . . . 4 (Rel 𝐴𝐴 = 𝐴)
3 dfrel2 5724 . . . . . . 7 (Rel 𝐵𝐵 = 𝐵)
4 cnveq 5434 . . . . . . . . 9 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqeq2 2781 . . . . . . . . 9 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
64, 5syl5ibr 236 . . . . . . . 8 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
76eqcoms 2778 . . . . . . 7 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
83, 7sylbi 207 . . . . . 6 (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
9 eqeq1 2774 . . . . . . 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 2778 . . . 4 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
132, 12sylbi 207 . . 3 (Rel 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1413imp 393 . 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 382   = wceq 1630  ccnv 5248  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257
This theorem is referenced by:  cnveq0  5732  weisoeq2  6748  relexpaddg  14000  relexpaddss  38529
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