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

Theorem csbieb 3174
 Description: Bidirectional conversion between an implicit class substitution hypothesis x = A → B = C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
Hypotheses
Ref Expression
csbieb.1 A V
csbieb.2 xC
Assertion
Ref Expression
csbieb (x(x = AB = C) ↔ [A / x]B = C)
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   C(x)

Proof of Theorem csbieb
StepHypRef Expression
1 csbieb.1 . 2 A V
2 csbieb.2 . 2 xC
3 csbiebt 3172 . 2 ((A V xC) → (x(x = AB = C) ↔ [A / x]B = C))
41, 2, 3mp2an 653 1 (x(x = AB = C) ↔ [A / x]B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-3an 936  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-sbc 3047  df-csb 3137 This theorem is referenced by:  csbiebg  3175
 Copyright terms: Public domain W3C validator