Theorem exanali 1859

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)

Assertion

Ref	Expression
exanali	⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))

Proof of Theorem exanali

Step	Hyp	Ref	Expression
1		annim 406	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
2	1	exbii 1848	. 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ¬ (𝜑 → 𝜓))
3		exnal 1827	. 2 ⊢ (∃𝑥 ¬ (𝜑 → 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))
4	2, 3	bitri 277	1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  sbnvOLD  2322  sbnOLD  2541  sbnALT  2595  gencbval  3551  dfss6  3957  nss  4029  nssss  5348  brprcneu  6662  marypha1lem  8897  reclem2pr  10470  dftr6  32986  brsset  33350  dfon3  33353  dffun10  33375  elfuns  33376  ecinn0  35622  ax12indn  36094  expandrexn  40647  vk15.4j  40882  vk15.4jVD  41268