Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
||
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 34882) 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 34877 | . . 3 ⊢ (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
2 | 1 | notbii 322 | . 2 ⊢ (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) |
3 | df-wl-rex 34882 | . 2 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑) | |
4 | exnalimn 1843 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
5 | 2, 3, 4 | 3bitr4i 305 | 1 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃wex 1779 ∈ wcel 2113 ∀wl-ral 34867 ∃wl-rex 34868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-11 2160 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-clel 2892 df-wl-ral 34872 df-wl-rex 34882 |
This theorem is referenced by: wl-dfreuv 34894 |
Copyright terms: Public domain | W3C validator |