Theorem exnalim 1534

Description: One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)

Assertion

Ref	Expression
exnalim	⊢ (∃x ¬ φ → ¬ ∀xφ)

Proof of Theorem exnalim

Step	Hyp	Ref	Expression
1		alexim 1533	. 2 ⊢ (∀xφ → ¬ ∃x ¬ φ)
2	1	con2i 557	1 ⊢ (∃x ¬ φ → ¬ ∀xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1240  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347

This theorem is referenced by:  exanaliim  1535  alexnim  1536  dtru  4238  brprcneu  5114