Theorem exanaliim 1520

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 1473	. 2 ⊢ (∃x(φ ∧ ¬ ψ) → ∃x ¬ (φ → ψ))
3		exnalim 1519	. 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 1226  ∃wex 1362

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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330

This theorem is referenced by:  rexnalim  2293  nssr  2978  nssssr  3930  brprcneu  5094