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Mirrors > Home > MPE Home > Th. List > ralnex2 | Structured version Visualization version GIF version |
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 3070 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜑) | |
2 | 1 | ralbii 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐵 𝜑) |
3 | ralnex 3070 | . 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 3059 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-ral 3060 df-rex 3069 |
This theorem is referenced by: ralnex3 3132 r2exlem 3141 rexcom 3288 genpnnp 11043 axtgupdim2 28494 uhgrvd00 29567 nrt2irr 30502 ply1dg3rt0irred 33587 dff15 35077 fmlaomn0 35375 gonan0 35377 goaln0 35378 hashnexinj 42110 fourierdlem42 46105 ichnreuop 47397 |
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