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Mathbox for Glauco Siliprandi |
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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 4346 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) |
3 | biimt 359 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) | |
4 | 2, 3 | sylbi 216 | . 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) |
5 | df-ral 3059 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
6 | r19.3rzf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
7 | 6 | 19.23 2199 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
8 | 5, 7 | bitri 274 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
9 | 4, 8 | bitr4di 288 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2879 ≠ wne 2937 ∀wral 3058 ∅c0 4326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-dif 3952 df-nul 4327 |
This theorem is referenced by: r19.28zf 44560 |
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