Theorem alinexa 1535

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 657	. . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2	1	albii 1400	. 2 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))
3		alnex 1429	. 2 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
4	2, 3	bitri 182	1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1283  ∃wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291

This theorem is referenced by:  sbnv  1811  ralnex  2363