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| Mirrors > Home > MPE Home > Th. List > Mathboxes > r19.3rzf | Structured version Visualization version GIF version | ||
| Description: Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
| Ref | Expression |
|---|---|
| r19.3rzf.1 | ⊢ Ⅎ𝑥𝜑 |
| r19.3rzf.2 | ⊢ Ⅎ𝑥𝐴 |
| Ref | Expression |
|---|---|
| r19.3rzf | ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.3rzf.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | 1 | n0f 4349 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
| 3 | biimt 360 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) | |
| 4 | 2, 3 | sylbi 217 | . 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) |
| 5 | df-ral 3062 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
| 6 | r19.3rzf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 7 | 6 | 19.23 2211 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
| 8 | 5, 7 | bitri 275 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
| 9 | 4, 8 | bitr4di 289 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 ∀wral 3061 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: r19.28zf 45164 |
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