Theorem alsc2d 44888

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 → ∀!𝑥 ∈ 𝐴𝜓)

Assertion

Ref	Expression
alsc2d	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Proof of Theorem alsc2d

Step	Hyp	Ref	Expression
1		alsc2d.1	. . 3 ⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓)
2		df-alsc 44883	. . 3 ⊢ (∀!𝑥 ∈ 𝐴𝜓 ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 𝑥 ∈ 𝐴))
3	1, 2	sylib 220	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 𝑥 ∈ 𝐴))
4	3	simprd 498	1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  ∀wral 3138  ∀!walsc 44881

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 209  df-an 399  df-alsc 44883

This theorem is referenced by:  alscn0d  44889