Theorem exanaliim 1661

Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)

Assertion

Ref	Expression
exanaliim	⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓))

Proof of Theorem exanaliim

Step	Hyp	Ref	Expression
1		annimim 687	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	eximi 1614	. 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑 → 𝜓))
3		exnalim 1660	. 2 ⊢ (∃𝑥 ¬ (𝜑 → 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓))
4	2, 3	syl 14	1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1506

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475

This theorem is referenced by:  rexnalim  2486  nssr  3244  nssssr  4256  brprcneu  5554