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Mirrors > Home > MPE Home > Th. List > Mathboxes > exellim | Structured version Visualization version GIF version |
Description: Closed form of exellimddv 34508. See also exlimim 34505 for a more general theorem. (Contributed by ML, 17-Jul-2020.) |
Ref | Expression |
---|---|
exellim | ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2146 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑) | |
2 | nfv 1906 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | sp 2172 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑)) | |
4 | 1, 2, 3 | exlimd 2208 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
5 | 4 | impcom 408 | 1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∈ wcel 2105 |
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 |
This theorem is referenced by: exellimddv 34508 |
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