Theorem ralnex2 2690

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 2538	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜑)
2	1	ralbii 2556	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑)
3		ralnex 2538	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 184	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wral 2528  ∃wrex 2529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-4 1563  ax-17 1579

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is referenced by: (None)