Theorem ralnex2 3114

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 3056	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜑)
2	1	ralbii 3076	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑)
3		ralnex 3056	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wral 3045  ∃wrex 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-ral 3046  df-rex 3055

This theorem is referenced by:  ralnex3  3115  r2exlem  3123  rexcom  3267  genpnnp  10965  axtgupdim2  28405  uhgrvd00  29469  nrt2irr  30409  ply1dg3rt0irred  33558  dff15  35081  fmlaomn0  35384  gonan0  35386  goaln0  35387  hashnexinj  42123  fourierdlem42  46154  ichnreuop  47477