Theorem als1d 50628

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
als1d.1	⊢ (𝜑 → ∀∃𝑥(𝜓 → 𝜒))

Assertion

Ref	Expression
als1d	⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))

Proof of Theorem als1d

Step	Hyp	Ref	Expression
1		als1d.1	. . 3 ⊢ (𝜑 → ∀∃𝑥(𝜓 → 𝜒))
2		df-als 50623	. . 3 ⊢ (∀∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝜒) ∧ ∃𝑥𝜓))
4	3	simpld 500	1 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812  ∀∃wals 50621

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-als 50623

This theorem is used by: (None)