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Theorem alsc1d 11280
Description: Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
Hypothesis
Ref Expression
alsc1d.1 (𝜑 → ∀!𝑥𝐴𝜓)
Assertion
Ref Expression
alsc1d (𝜑 → ∀𝑥𝐴 𝜓)

Proof of Theorem alsc1d
StepHypRef Expression
1 alsc1d.1 . . 3 (𝜑 → ∀!𝑥𝐴𝜓)
2 df-alsc 11276 . . 3 (∀!𝑥𝐴𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∃𝑥 𝑥𝐴))
31, 2sylib 120 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ∧ ∃𝑥 𝑥𝐴))
43simpld 110 1 (𝜑 → ∀𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1424  wcel 1436  wral 2355  ∀!walsc 11274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104
This theorem depends on definitions:  df-bi 115  df-alsc 11276
This theorem is referenced by: (None)
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