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Mirrors > Home > ILE Home > Th. List > Mathboxes > alsi1d | GIF version |
Description: Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
Ref | Expression |
---|---|
alsi1d.1 | ⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) |
Ref | Expression |
---|---|
alsi1d | ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alsi1d.1 | . . 3 ⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒)) | |
2 | df-alsi 13646 | . . 3 ⊢ (∀!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓)) | |
3 | 1, 2 | sylib 121 | . 2 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓)) |
4 | 3 | simpld 111 | 1 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1333 ∃wex 1472 ∀!walsi 13644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 |
This theorem depends on definitions: df-bi 116 df-alsi 13646 |
This theorem is referenced by: (None) |
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