Theorem alinexa 1863

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)

Assertion

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

Proof of Theorem alinexa

Step	Hyp	Ref	Expression
1		imnang 1862	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))
2		alnex 1801	. 2 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
3	1, 2	bitri 277	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800

This theorem is referenced by:  exnalimn  1864  equsexvw  2025  sbn  2314  ceqsex  3501  ceqsexv  3502  zfregs2  9688  ac6n  10442  nnunb  12477  alexsubALTlem3  24109  nmobndseqi  30982  regsfromsetind  36899  bj-equsexvwd  37248  difunieq  37868  frege124d  44337  zfregs2VD  45416