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

Theorem nfcjust 2213
Description: Justification theorem for df-nfc 2214. (Contributed by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
nfcjust (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcjust
StepHypRef Expression
1 nfv 1464 . . 3 𝑥 𝑦 = 𝑧
2 eleq1 2147 . . 3 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
31, 2nfbidf 1475 . 2 (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑧𝐴))
43cbvalv 1839 1 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1285  wnf 1392  wcel 1436
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-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-cleq 2078  df-clel 2081
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator