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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 4159 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | biimt 352 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) | |
3 | 1, 2 | sylbi 209 | . 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) |
4 | df-ral 3095 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
5 | r19.3rz.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
6 | 5 | 19.23 2197 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
7 | 4, 6 | bitri 267 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
8 | 3, 7 | syl6bbr 281 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1599 ∃wex 1823 Ⅎwnf 1827 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∅c0 4141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-dif 3795 df-nul 4142 |
This theorem is referenced by: r19.28z 4286 r19.3rzv 4287 r19.27z 4293 2reu4a 42160 |
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