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Theorem alsi2d 11060
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.)
Hypothesis
Ref Expression
alsi2d.1 (𝜑 → ∀!𝑥(𝜓𝜒))
Assertion
Ref Expression
alsi2d (𝜑 → ∃𝑥𝜓)

Proof of Theorem alsi2d
StepHypRef Expression
1 alsi2d.1 . . 3 (𝜑 → ∀!𝑥(𝜓𝜒))
2 df-alsi 11056 . . 3 (∀!𝑥(𝜓𝜒) ↔ (∀𝑥(𝜓𝜒) ∧ ∃𝑥𝜓))
31, 2sylib 120 . 2 (𝜑 → (∀𝑥(𝜓𝜒) ∧ ∃𝑥𝜓))
43simprd 112 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283  wex 1422  ∀!walsi 11054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105
This theorem depends on definitions:  df-bi 115  df-alsi 11056
This theorem is referenced by: (None)
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