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Mirrors > Home > MPE Home > Th. List > ralnex3 | Structured version Visualization version GIF version |
Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof shortened by Wolf Lammen, 18-May-2023.) |
Ref | Expression |
---|---|
ralnex3 | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralnex 3236 | . . 3 ⊢ (∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑧 ∈ 𝐶 𝜑) | |
2 | 1 | 2ralbii 3166 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ ∃𝑧 ∈ 𝐶 𝜑) |
3 | ralnex2 3260 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ ∃𝑧 ∈ 𝐶 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑) | |
4 | 2, 3 | bitri 277 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∀wral 3138 ∃wrex 3139 |
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 209 df-an 399 df-ex 1777 df-ral 3143 df-rex 3144 |
This theorem is referenced by: axtgupdim2 26251 |
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