Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dfreusb | Structured version Visualization version GIF version |
Description: An alternate definition of restricted existential uniqueness (df-wl-reu 34737) using substitution. (Contributed by Wolf Lammen, 28-May-2023.) |
Ref | Expression |
---|---|
wl-dfreusb | ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-dfrexsb 34732 | . . 3 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)) | |
2 | wl-dfrmosb 34734 | . . 3 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | anbi12i 626 | . 2 ⊢ ((∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑) ↔ (∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ∧ ∃*𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))) |
4 | df-wl-reu 34737 | . 2 ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ (∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑)) | |
5 | df-eu 2647 | . 2 ⊢ (∃!𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ∧ ∃*𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))) | |
6 | 3, 4, 5 | 3bitr4i 304 | 1 ⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 [wsb 2060 ∈ wcel 2105 ∃*wmo 2613 ∃!weu 2646 ∃wl-rex 34713 ∃*wl-rmo 34714 ∃!wl-reu 34715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-wl-ral 34717 df-wl-rex 34727 df-wl-rmo 34733 df-wl-reu 34737 |
This theorem is referenced by: (None) |
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