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Mirrors > Home > MPE Home > Th. List > r19.3rz | Structured version Visualization version GIF version |
Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.) |
Ref | Expression |
---|---|
r19.3rz.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
r19.3rz | ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4247 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | biimt 364 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) | |
3 | 1, 2 | sylbi 220 | . 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) |
4 | df-ral 3056 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
5 | r19.3rz.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
6 | 5 | 19.23 2211 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
7 | 4, 6 | bitri 278 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
8 | 3, 7 | bitr4di 292 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 ∃wex 1787 Ⅎwnf 1791 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-ne 2933 df-ral 3056 df-dif 3856 df-nul 4224 |
This theorem is referenced by: r19.28z 4395 r19.3rzv 4396 r19.27z 4402 2reu4lem 4423 |
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