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Theorem cnveqb 4699
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 4432 . 2 (A = BA = B)
2 dfrel2 4694 . . . 4 (Rel AA = A)
3 dfrel2 4694 . . . . . . 7 (Rel BB = B)
4 cnveq 4432 . . . . . . . . 9 (A = BA = B)
5 eqeq2 2027 . . . . . . . . 9 (B = B → (A = BA = B))
64, 5syl5ibr 145 . . . . . . . 8 (B = B → (A = BA = B))
76eqcoms 2021 . . . . . . 7 (B = B → (A = BA = B))
83, 7sylbi 114 . . . . . 6 (Rel B → (A = BA = B))
9 eqeq1 2024 . . . . . . 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 2021 . . . 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 1226  ccnv 4267  Rel wrel 4273
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276
This theorem is referenced by:  cnveq0  4700
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