ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  a16nf GIF version

Theorem a16nf 1877
Description: If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)
Assertion
Ref Expression
a16nf (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem a16nf
StepHypRef Expression
1 nfae 1730 . 2 𝑧𝑥 𝑥 = 𝑦
2 a16g 1875 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
31, 2nfd 1534 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wnf 1471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  nfsbxy  1954  nfsbxyt  1955  dvelimor  2030
  Copyright terms: Public domain W3C validator