Theorem alinexa 1945

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 1944	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))
2		alnex 1882	. 2 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
3	1, 2	bitri 267	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1656  ∃wex 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881

This theorem is referenced by:  equs3  2064  r2exlem  3269  zfregs2  8886  ac6n  9622  nnunb  11614  alexsubALTlem3  22223  nmobndseqi  28189  bj-exnalimn  33139  bj-ssbn  33177  frege124d  38894  zfregs2VD  39895