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Mirrors > Home > MPE Home > Th. List > Mathboxes > exellim | Structured version Visualization version GIF version |
Description: Closed form of exellimddv 34762. See also exlimim 34759 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 2152 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑) | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | sp 2180 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑)) | |
4 | 1, 2, 3 | exlimd 2216 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
5 | 4 | impcom 411 | 1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 ∃wex 1781 ∈ wcel 2111 |
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-10 2142 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 |
This theorem is referenced by: exellimddv 34762 |
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