Theorem exanaliim 1535

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 781	. . 3 ⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ))
2	1	eximi 1488	. 2 ⊢ (∃x(φ ∧ ¬ ψ) → ∃x ¬ (φ → ψ))
3		exnalim 1534	. 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 97  ∀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-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347

This theorem is referenced by:  rexnalim  2311  nssr  2997  nssssr  3948  brprcneu  5112