Theorem alsc2d

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)

	Ref	Expression
Hypothesis	alsc2d.1	⊢ ( 𝜑 → ∀! 𝑥 ∈ 𝐴 𝜓 )
Assertion	alsc2d	⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ 𝐴 )

Proof

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