Theorem alsi2d 10121

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.)

Hypothesis

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

Assertion

Ref	Expression
alsi2d	⊢ (𝜑 → ∃𝑥𝜓)

Proof of Theorem alsi2d

Step	Hyp	Ref	Expression
1		alsi2d.1	. . 3 ⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒))
2		df-alsi 10117	. . 3 ⊢ (∀!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓))
3	1, 2	sylib 127	. 2 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓))
4	3	simprd 107	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381  ∀!walsi 10115

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100

This theorem depends on definitions:  df-bi 110  df-alsi 10117

This theorem is referenced by: (None)