Theorem exalim 1500

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 1499. (Contributed by Jim Kingdon, 29-Jul-2018.)

Assertion

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

Proof of Theorem exalim

Step	Hyp	Ref	Expression
1		alnex 1497	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2	1	biimpi 120	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
3	2	con2i 627	1 ⊢ (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1351  ∃wex 1490

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 1445  ax-gen 1447  ax-ie2 1492

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

This theorem is referenced by:  n0rf  3433  ax9vsep  4121