Theorem ralinexa 3183

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Assertion

Ref	Expression
ralinexa	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Proof of Theorem ralinexa

Step	Hyp	Ref	Expression
1		imnan 403	. . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2	1	ralbii 3088	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓))
3		ralnex 3158	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
4	2, 3	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wral 3061  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-ral 3066  df-rex 3067

This theorem is referenced by:  kmlem7  9770  kmlem13  9776  lspsncv0  20183  ntreq0  21974  lhop1lem  24910  soseq  33540  nogt01o  33636  ltrnnid  37887