New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  cnveq GIF version

Theorem cnveq 4886
 Description: Equality theorem for converse. (Contributed by set.mm contributors, 13-Aug-1995.)
Assertion
Ref Expression
cnveq (A = BA = B)

Proof of Theorem cnveq
StepHypRef Expression
1 cnvss 4885 . . 3 (A BA B)
2 cnvss 4885 . . 3 (B AB A)
31, 2anim12i 549 . 2 ((A B B A) → (A B B A))
4 eqss 3287 . 2 (A = B ↔ (A B B A))
5 eqss 3287 . 2 (A = B ↔ (A B B A))
63, 4, 53imtr4i 257 1 (A = BA = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3257  ◡ccnv 4771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-cnv 4785 This theorem is referenced by:  cnveqi  4887  cnveqd  4888  cnveqb  5063  funcnvuni  5161  f1eq1  5253  f1o00  5317  enprmaplem3  6078  enprmaplem5  6080  enprmaplem6  6081
 Copyright terms: Public domain W3C validator