 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  a16nf GIF version

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
StepHypRef Expression
1 nfae 1954 . 2 zx x = y
2 a16g 1945 . 2 (x x = y → (φzφ))
31, 2nfd 1766 1 (x x = y → Ⅎzφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
 Copyright terms: Public domain W3C validator