Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnveq GIF version

Theorem cnveq 4708
 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 4707 . . 3 (𝐴𝐵𝐴𝐵)
2 cnvss 4707 . . 3 (𝐵𝐴𝐵𝐴)
31, 2anim12i 336 . 2 ((𝐴𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))
4 eqss 3107 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3107 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
63, 4, 53imtr4i 200 1 (𝐴 = 𝐵𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ⊆ wss 3066  ◡ccnv 4533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-br 3925  df-opab 3985  df-cnv 4542 This theorem is referenced by:  cnveqi  4709  cnveqd  4710  rneq  4761  cnveqb  4989  funcnvuni  5187  f1eq1  5318  f1o00  5395  foeqcnvco  5684  tposfn2  6156  ereq1  6429  infeq3  6895  iscn  12355  ishmeo  12462
 Copyright terms: Public domain W3C validator