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 403	. . 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 275	1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  gencbval  3498  dfss6  3925  nss  4000  nssss  5398  brprcneu  6812  brprcneuALT  6813  marypha1lem  9323  reclem2pr  10942  dftr6  35744  brsset  35883  dfon3  35886  dffun10  35908  elfuns  35909  ecinn0  38341  ax12indn  38942  expandrexn  44284  vk15.4j  44522  vk15.4jVD  44907