Theorem alinexa 1494

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		imnan 624	. . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2	1	albii 1359	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))
3		alnex 1388	. 2 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
4	2, 3	bitri 173	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  sbnv  1768  ralnex  2316