Theorem exanali 1786

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 441	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
2	1	exbii 1774	. 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ¬ (𝜑 → 𝜓))
3		exnal 1754	. 2 ⊢ (∃𝑥 ¬ (𝜑 → 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))
4	2, 3	bitri 264	1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  sbn  2391  gencbval  3252  dfss6  3593  nss  3663  nssss  4924  brprcneu  6184  marypha1lem  8339  reclem2pr  9870  dftr6  31640  brsset  31996  dfon3  31999  dffun10  32021  elfuns  32022  ecinn0  34118  ax12indn  34228  vk15.4j  38734  vk15.4jVD  39150