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Theorem cnveqb 4719
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 4452 . 2 (A = BA = B)
2 dfrel2 4714 . . . 4 (Rel AA = A)
3 dfrel2 4714 . . . . . . 7 (Rel BB = B)
4 cnveq 4452 . . . . . . . . 9 (A = BA = B)
5 eqeq2 2046 . . . . . . . . 9 (B = B → (A = BA = B))
64, 5syl5ibr 145 . . . . . . . 8 (B = B → (A = BA = B))
76eqcoms 2040 . . . . . . 7 (B = B → (A = BA = B))
83, 7sylbi 114 . . . . . 6 (Rel B → (A = BA = B))
9 eqeq1 2043 . . . . . . 7 (A = A → (A = BA = B))
109imbi2d 219 . . . . . 6 (A = A → ((A = BA = B) ↔ (A = BA = B)))
118, 10syl5ibr 145 . . . . 5 (A = A → (Rel B → (A = BA = B)))
1211eqcoms 2040 . . . 4 (A = A → (Rel B → (A = BA = B)))
132, 12sylbi 114 . . 3 (Rel A → (Rel B → (A = BA = B)))
1413imp 115 . 2 ((Rel A Rel B) → (A = BA = B))
151, 14impbid2 131 1 ((Rel A Rel B) → (A = BA = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  ccnv 4287  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  cnveq0  4720
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