Theorem exalim 1490

Description: One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1489. (Contributed by Jim Kingdon, 29-Jul-2018.)

Assertion

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

Proof of Theorem exalim

Step	Hyp	Ref	Expression
1		alnex 1487	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2	1	biimpi 119	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
3	2	con2i 617	1 ⊢ (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1341  ∃wex 1480

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 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie2 1482

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349

This theorem is referenced by:  n0rf  3421  ax9vsep  4105