Theorem exnalim 1521

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 1520	. 2 ⊢ (∀xφ → ¬ ∃x ¬ φ)
2	1	con2i 545	1 ⊢ (∃x ¬ φ → ¬ ∀xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1226  ∃wex 1363

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 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-4 1382  ax-17 1401  ax-ial 1410

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330

This theorem is referenced by:  exanaliim  1522  alexnim  1523  dtru  4220  brprcneu  5094