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Mirrors > Home > MPE Home > Th. List > ralnex2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralnex2 3257 as of 18-May-2023. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralnex2OLD | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 316 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑) | |
2 | notnotb 316 | . . . 4 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
3 | 2 | 2rexbii 3245 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ ¬ 𝜑) |
4 | rexnal2 3255 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑) | |
5 | 3, 4 | bitr2i 277 | . 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
6 | 1, 5 | xchbinx 335 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∀wral 3135 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-ral 3140 df-rex 3141 |
This theorem is referenced by: (None) |
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