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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dfrexv | Structured version Visualization version GIF version |
Description: Alternate definition of restricted universal quantification (df-wl-rex 35011) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 25-May-2023.) |
Ref | Expression |
---|---|
wl-dfrexv | ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-dfralv 35006 | . . 3 ⊢ (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
2 | 1 | notbii 323 | . 2 ⊢ (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) |
3 | df-wl-rex 35011 | . 2 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑) | |
4 | exnalimn 1845 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
5 | 2, 3, 4 | 3bitr4i 306 | 1 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 ∈ wcel 2111 ∀wl-ral 34996 ∃wl-rex 34997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-11 2158 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-clel 2870 df-wl-ral 35001 df-wl-rex 35011 |
This theorem is referenced by: wl-dfreuv 35023 |
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