Theorem a16nf 1914

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 1767	. 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
2		a16g 1912	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
3	1, 2	nfd 1572	1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Syntax hints:   → wi 4  ∀wal 1396  Ⅎwnf 1509

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811

This theorem is referenced by:  nfsbxy  1995  nfsbxyt  1996  dvelimor  2071