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Mirrors > Home > MPE Home > Th. List > ralnex3OLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralnex3 3262 as of 18-May-2023. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralnex3OLD | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 317 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑) | |
2 | notnotb 317 | . . . . 5 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
3 | 2 | rexbii 3247 | . . . 4 ⊢ (∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ¬ ¬ 𝜑) |
4 | 3 | 2rexbii 3248 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ¬ ¬ 𝜑) |
5 | rexnal3 3259 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑) | |
6 | 4, 5 | bitr2i 278 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑) |
7 | 1, 6 | xchbinx 336 | 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 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-ral 3143 df-rex 3144 |
This theorem is referenced by: (None) |
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