Theorem exanaliim 1590

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 823	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	eximi 1543	. 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑 → 𝜓))
3		exnalim 1589	. 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 103  ∀wal 1294  ∃wex 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402

This theorem is referenced by:  rexnalim  2381  nssr  3099  nssssr  4073  brprcneu  5333