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Theorem a16nf 1794
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 1654 . 2 𝑧𝑥 𝑥 = 𝑦
2 a16g 1792 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
31, 2nfd 1461 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wnf 1394
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  nfsbxy  1866  nfsbxyt  1867  dvelimor  1942
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