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| Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.) | 
| Ref | Expression | 
|---|---|
| ralnex2 | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ralnex 3072 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜑) | |
| 2 | 1 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑) | 
| 3 | ralnex 3072 | . 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wral 3061 ∃wrex 3070 | 
| 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 3062 df-rex 3071 | 
| This theorem is referenced by: ralnex3 3134 r2exlem 3143 rexcom 3290 genpnnp 11045 axtgupdim2 28479 uhgrvd00 29552 nrt2irr 30492 ply1dg3rt0irred 33607 dff15 35098 fmlaomn0 35395 gonan0 35397 goaln0 35398 hashnexinj 42129 fourierdlem42 46164 ichnreuop 47459 | 
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