Theorem exnalim 1660

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 1659	. 2 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
2	1	con2i 628	1 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475

This theorem is referenced by:  exanaliim  1661  alexnim  1662  nnal  1663  dtru  4596  brprcneu  5551