Theorem exalim 1431

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 1430. (Contributed by Jim Kingdon, 29-Jul-2018.)

Assertion

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

Proof of Theorem exalim

Step	Hyp	Ref	Expression
1		alnex 1428	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2	1	biimpi 118	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
3	2	con2i 589	1 ⊢ (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1282  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  n0rf  3260  ax9vsep  3901