Theorem exnalim 1646

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 1645	. 2 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
2	1	con2i 627	1 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1351  ∃wex 1492

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461

This theorem is referenced by:  exanaliim  1647  alexnim  1648  nnal  1649  dtru  4558  brprcneu  5507