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Mirrors > Home > ILE Home > Th. List > ralinexa | 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 662 | . . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
2 | 1 | ralbii 2413 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓)) |
3 | ralnex 2398 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∀wral 2388 ∃wrex 2389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-5 1404 ax-gen 1406 ax-ie2 1451 ax-4 1468 ax-17 1487 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-fal 1318 df-nf 1418 df-ral 2393 df-rex 2394 |
This theorem is referenced by: ntreq0 12138 |
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