Theorem a16nf 1866

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

Step	Hyp	Ref	Expression
1		nfae 1719	. 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
2		a16g 1864	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
3	1, 2	nfd 1523	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Syntax hints:   → wi 4  ∀wal 1351  Ⅎwnf 1460

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  nfsbxy  1942  nfsbxyt  1943  dvelimor  2018