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Theorem alsc2d 13639
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 13634 . . 3 (∀!𝑥𝐴𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∃𝑥 𝑥𝐴))
31, 2sylib 121 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ∧ ∃𝑥 𝑥𝐴))
43simprd 113 1 (𝜑 → ∃𝑥 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1472  wcel 2128  wral 2435  ∀!walsc 13632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106
This theorem depends on definitions:  df-bi 116  df-alsc 13634
This theorem is referenced by: (None)
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