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

Theorem cnveq 4537
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 4536 . . 3 (𝐴𝐵𝐴𝐵)
2 cnvss 4536 . . 3 (𝐵𝐴𝐵𝐴)
31, 2anim12i 331 . 2 ((𝐴𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))
4 eqss 3015 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3015 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
63, 4, 53imtr4i 199 1 (𝐴 = 𝐵𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wss 2974  ccnv 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-br 3794  df-opab 3848  df-cnv 4379
This theorem is referenced by:  cnveqi  4538  cnveqd  4539  rneq  4589  cnveqb  4806  funcnvuni  4999  f1eq1  5118  f1o00  5192  foeqcnvco  5461  tposfn2  5915  ereq1  6179  infeq3  6487
  Copyright terms: Public domain W3C validator