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Theorem a16nf 1846
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 1699 . 2 𝑧𝑥 𝑥 = 𝑦
2 a16g 1844 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
31, 2nfd 1503 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333  wnf 1440
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743
This theorem is referenced by:  nfsbxy  1922  nfsbxyt  1923  dvelimor  1998
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