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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 4309 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | biimt 363 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) | |
3 | 1, 2 | sylbi 219 | . 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) |
4 | df-ral 3143 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
5 | r19.3rz.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
6 | 5 | 19.23 2207 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
7 | 4, 6 | bitri 277 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
8 | 3, 7 | syl6bbr 291 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∅c0 4290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-dif 3938 df-nul 4291 |
This theorem is referenced by: r19.28z 4442 r19.3rzv 4443 r19.27z 4449 2reu4lem 4464 |
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