Theorem a16nf 2051

Description: If dtru in set.mm is false, then there is only one element in the universe, so 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 1954	. 2 ⊢ Ⅎz∀x x = y
2		a16g 1945	. 2 ⊢ (∀x x = y → (φ → ∀zφ))
3	1, 2	nfd 1766	1 ⊢ (∀x x = y → Ⅎzφ)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfsb  2109  nfsbd  2111