Theorem exnalim 1625

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 1624	. 2 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
2	1	con2i 616	1 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437

This theorem is referenced by:  exanaliim  1626  alexnim  1627  dtru  4470  brprcneu  5407  bj-nnal  12938