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

Theorem csbieb 3047
 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 / 𝑥𝐵 = 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 3045 . 2 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
41, 2, 3mp2an 423 1 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332   ∈ wcel 1481  Ⅎwnfc 2269  Vcvv 2690  ⦋csb 3008 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-sbc 2915  df-csb 3009 This theorem is referenced by:  csbiebg  3048
 Copyright terms: Public domain W3C validator