Theorem alsi2d

Description: Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018)

	Ref	Expression
Hypothesis	alsi2d.1	⊢ ( 𝜑 → ∀! 𝑥 ( 𝜓 → 𝜒 ) )
Assertion	alsi2d	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		alsi2d.1	⊢ ( 𝜑 → ∀! 𝑥 ( 𝜓 → 𝜒 ) )
2		df-alsi	⊢ ( ∀! 𝑥 ( 𝜓 → 𝜒 ) ↔ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) ∧ ∃ 𝑥 𝜓 ) )
3	1 2	sylib	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) ∧ ∃ 𝑥 𝜓 ) )
4	3	simprd	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )