Theorem exanaliim 1671

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 688	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	eximi 1624	. 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑 → 𝜓))
3		exnalim 1670	. 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 1371  ∃wex 1516

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485

This theorem is referenced by:  rexnalim  2496  nssr  3257  nssssr  4274  brprcneu  5582