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| Mirrors > Home > MPE Home > Th. List > ralinexa | Structured version Visualization version GIF version | ||
| Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) |
| Ref | Expression |
|---|---|
| ralinexa | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imnan 399 | . . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
| 2 | 1 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓)) |
| 3 | ralnex 3072 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀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: soseq 8184 kmlem7 10197 kmlem13 10203 lspsncv0 21148 ntreq0 23085 lhop1lem 26052 nogt01o 27741 ltrnnid 40138 |
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