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Mirrors > Home > MPE Home > Th. List > Mathboxes > fipjust | Structured version Visualization version GIF version |
Description: A definition of the finite intersection property of a class based on closure under pairwise intersection of its elements is independent of the dummy variables. (Contributed by RP, 1-Jan-2020.) |
Ref | Expression |
---|---|
fipjust | ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 ∩ 𝑣) ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 4183 | . . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∩ 𝑣) = (𝑥 ∩ 𝑣)) | |
2 | 1 | eleq1d 2899 | . 2 ⊢ (𝑢 = 𝑥 → ((𝑢 ∩ 𝑣) ∈ 𝐴 ↔ (𝑥 ∩ 𝑣) ∈ 𝐴)) |
3 | ineq2 4185 | . . 3 ⊢ (𝑣 = 𝑦 → (𝑥 ∩ 𝑣) = (𝑥 ∩ 𝑦)) | |
4 | 3 | eleq1d 2899 | . 2 ⊢ (𝑣 = 𝑦 → ((𝑥 ∩ 𝑣) ∈ 𝐴 ↔ (𝑥 ∩ 𝑦) ∈ 𝐴)) |
5 | 2, 4 | cbvral2vw 3463 | 1 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 ∩ 𝑣) ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 ∀wral 3140 ∩ cin 3937 |
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-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ral 3145 df-rab 3149 df-in 3945 |
This theorem is referenced by: (None) |
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