Theorem exanali 1858

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 403	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
2	1	exbii 1846	. 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ¬ (𝜑 → 𝜓))
3		exnal 1825	. 2 ⊢ (∃𝑥 ¬ (𝜑 → 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))
4	2, 3	bitri 275	1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778

This theorem is referenced by:  gencbval  3555  dfss6  3998  nss  4073  nssss  5475  brprcneu  6910  brprcneuALT  6911  marypha1lem  9502  reclem2pr  11117  dftr6  35713  brsset  35853  dfon3  35856  dffun10  35878  elfuns  35879  ecinn0  38309  ax12indn  38899  expandrexn  44260  vk15.4j  44499  vk15.4jVD  44885