Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc16nf Structured version   Visualization version   Unicode version

Theorem axc16nf 2027
 Description: If dtru 4594 is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 1920. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.)
Assertion
Ref Expression
axc16nf
Distinct variable group:   ,
Allowed substitution hints:   (,,)

Proof of Theorem axc16nf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 aev 2026 . 2
2 nfa1 1979 . . 3
3 axc16 2024 . . 3
42, 3nfd 1956 . 2
51, 4syl 17 1
 Colors of variables: wff setvar class Syntax hints:   wi 4  wal 1442  wnf 1667 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933 This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668 This theorem is referenced by:  nfsb  2269  nfsbd  2271
 Copyright terms: Public domain W3C validator