Theorem ralnex2 2571

Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)

Assertion

Ref	Expression
ralnex2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Proof of Theorem ralnex2

Step	Hyp	Ref	Expression
1		ralnex 2426	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜑)
2	1	ralbii 2441	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑)
3		ralnex 2426	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 183	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wral 2416  ∃wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-ral 2421  df-rex 2422

This theorem is referenced by: (None)