Theorem exnalimn 1843

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 1842	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
2	1	con2bii 360	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  ∃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 209  df-an 399  df-ex 1780

This theorem is referenced by:  r2exlem  3301  wl-dfrexf  34885  wl-dfrexv  34887