Theorem exnalimn 1863

Description: Existential quantification of a conjunction expressed with only primitive symbols (→, ¬, ∀). (Contributed by NM, 10-May-1993.) State the most general instance. (Revised by BJ, 29-Sep-2019.)

Assertion

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

Proof of Theorem exnalimn

Step	Hyp	Ref	Expression
1		alinexa 1862	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
2	1	con2bii 359	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  ax12ev2  2214  r2exlem  3150  regsfromsetind  36852  mh-prprimbi  36856  mh-infprim1bi  36859