Theorem n0alsm 17065

Description: If 𝐴 is inhabited, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)

Assertion

Ref	Expression
n0alsm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem n0alsm

Step	Hyp	Ref	Expression
1		alsraln0m 17062	. 2 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))
2	1	rbaib 933	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∀∃wals 17034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-ral 2533  df-als 17036

This theorem is referenced by: (None)