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

Theorem cnveq 4785
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 4784 . . 3 (𝐴𝐵𝐴𝐵)
2 cnvss 4784 . . 3 (𝐵𝐴𝐵𝐴)
31, 2anim12i 336 . 2 ((𝐴𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))
4 eqss 3162 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3162 . 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 1348  wss 3121  ccnv 4610
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3990  df-opab 4051  df-cnv 4619
This theorem is referenced by:  cnveqi  4786  cnveqd  4787  rneq  4838  cnveqb  5066  funcnvuni  5267  f1eq1  5398  f1o00  5477  foeqcnvco  5769  tposfn2  6245  ereq1  6520  infeq3  6992  1arith  12319  iscn  12991  ishmeo  13098
  Copyright terms: Public domain W3C validator