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Theorem cnveq 4452
Description: Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
Assertion
Ref Expression
cnveq (A = BA = B)

Proof of Theorem cnveq
StepHypRef Expression
1 cnvss 4451 . . 3 (ABAB)
2 cnvss 4451 . . 3 (BABA)
31, 2anim12i 321 . 2 ((AB BA) → (AB BA))
4 eqss 2954 . 2 (A = B ↔ (AB BA))
5 eqss 2954 . 2 (A = B ↔ (AB BA))
63, 4, 53imtr4i 190 1 (A = BA = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wss 2911  ccnv 4287
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-cnv 4296
This theorem is referenced by:  cnveqi  4453  cnveqd  4454  rneq  4504  cnveqb  4719  funcnvuni  4911  f1eq1  5030  f1o00  5104  foeqcnvco  5373  tposfn2  5822  ereq1  6049
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