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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dfrmov | Structured version Visualization version GIF version |
Description: Alternate definition of restricted "at most one" (df-wl-rmo 34867) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.) |
Ref | Expression |
---|---|
wl-dfrmov | ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-dfralv 34856 | . . . 4 ⊢ (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
2 | impexp 453 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
3 | 2 | albii 1820 | . . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) |
4 | 1, 3 | bitr4i 280 | . . 3 ⊢ (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)) |
5 | 4 | exbii 1848 | . 2 ⊢ (∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)) |
6 | df-wl-rmo 34867 | . 2 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦)) | |
7 | df-mo 2622 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)) | |
8 | 5, 6, 7 | 3bitr4i 305 | 1 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 ∈ wcel 2114 ∃*wmo 2620 ∀wl-ral 34846 ∃*wl-rmo 34848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-11 2161 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-mo 2622 df-clel 2893 df-wl-ral 34851 df-wl-rmo 34867 |
This theorem is referenced by: wl-dfreuv 34873 |
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