Theorem alsi1d 13957

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

Hypothesis

Ref	Expression
alsi1d.1	⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒))

Assertion

Ref	Expression
alsi1d	⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))

Proof of Theorem alsi1d

Step	Hyp	Ref	Expression
1		alsi1d.1	. . 3 ⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒))
2		df-alsi 13954	. . 3 ⊢ (∀!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓))
3	1, 2	sylib 121	. 2 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓))
4	3	simpld 111	1 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1341  ∃wex 1480  ∀!walsi 13952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-alsi 13954

This theorem is referenced by: (None)