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Theorem alsc2d 42349
 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.)
Hypothesis
Ref Expression
alsc2d.1 (𝜑 → ∀!𝑥𝐴𝜓)
Assertion
Ref Expression
alsc2d (𝜑 → ∃𝑥 𝑥𝐴)

Proof of Theorem alsc2d
StepHypRef Expression
1 alsc2d.1 . . 3 (𝜑 → ∀!𝑥𝐴𝜓)
2 df-alsc 42344 . . 3 (∀!𝑥𝐴𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∃𝑥 𝑥𝐴))
31, 2sylib 207 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ∧ ∃𝑥 𝑥𝐴))
43simprd 478 1 (𝜑 → ∃𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∀!walsc 42342 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 196  df-an 385  df-alsc 42344 This theorem is referenced by:  alscn0d  42350
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