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Theorem a16nf 1746
 Description: If there is only one element in the universe, then everything satisfies Ⅎ. (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 1607 . 2 𝑧𝑥 𝑥 = 𝑦
2 a16g 1744 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
31, 2nfd 1416 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  nfsbxy  1818  nfsbxyt  1819  dvelimor  1894
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