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Mirrors > Home > MPE Home > Th. List > Mathboxes > alsi2d | Structured version Visualization version GIF version |
Description: Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
Ref | Expression |
---|---|
alsi2d.1 | ⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) |
Ref | Expression |
---|---|
alsi2d | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alsi2d.1 | . . 3 ⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) | |
2 | df-alsi 44896 | . . 3 ⊢ (∀!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓)) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓)) |
4 | 3 | simprd 498 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 ∃wex 1780 ∀!walsi 44894 |
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 209 df-an 399 df-alsi 44896 |
This theorem is referenced by: (None) |
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