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 3060	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜑)
2	1	ralbii 3080	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑)
3		ralnex 3060	. 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 3049  ∃wrex 3058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-ral 3050  df-rex 3059

This theorem is referenced by:  ralnex3  3115  r2exlem  3123  rexcom  3263  genpnnp  10914  axtgupdim2  28492  uhgrvd00  29557  nrt2irr  30497  ply1dg3rt0irred  33614  dff15  35189  fmlaomn0  35533  gonan0  35535  goaln0  35536  hashnexinj  42321  fourierdlem42  46335  ichnreuop  47660