Theorem exalim 1388

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 1387. (Contributed by Jim Kingdon, 29-Jul-2018.)

Assertion

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

Proof of Theorem exalim

Step	Hyp	Ref	Expression
1		alnex 1385	. . 3 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
2	1	biimpi 113	. 2 ⊢ (∀x ¬ φ → ¬ ∃xφ)
3	2	con2i 557	1 ⊢ (∃xφ → ¬ ∀x ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1240  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie2 1380

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248

This theorem is referenced by:  n0rf  3227  ax9vsep  3871