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Mirrors > Home > MPE Home > Th. List > Mathboxes > alsc2d | Structured version Visualization version GIF version |
Description: Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
Ref | Expression |
---|---|
alsc2d.1 | ⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) |
Ref | Expression |
---|---|
alsc2d | ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alsc2d.1 | . . 3 ⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓) | |
2 | df-alsc 46192 | . . 3 ⊢ (∀!𝑥 ∈ 𝐴𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | sylib 221 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 𝑥 ∈ 𝐴)) |
4 | 3 | simprd 499 | 1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1787 ∈ wcel 2111 ∀wral 3062 ∀!walsc 46190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-alsc 46192 |
This theorem is referenced by: alscn0d 46198 |
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