Theorem exanaliim 1521

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 775	. . 3 ⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ))
2	1	eximi 1474	. 2 ⊢ (∃x(φ ∧ ¬ ψ) → ∃x ¬ (φ → ψ))
3		exnalim 1520	. 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 1314  ∃wex 1361

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 532  ax-in2 533  ax-5 1315  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-4 1382  ax-17 1401  ax-ial 1410

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1232  df-nf 1329

This theorem is referenced by:  rexnalim  2293  nssr  2981  nssssr  3910  brprcneu  5063