Theorem ralseu2d 17146

Description: Deduction rule: Given "all some one" applied to a class, you can extract the "exactly one" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 22-Jul-2026.)

Hypothesis

Ref	Expression
ralseu2d.1	⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))

Assertion

Ref	Expression
ralseu2d	⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

Proof of Theorem ralseu2d

Step	Hyp	Ref	Expression
1		ralseu2d.1	. . 3 ⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))
2		df-ralseu 17137	. . 3 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃!𝑥 ∈ 𝐴 𝜓))
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃!𝑥 ∈ 𝐴 𝜓))
4	3	simprd 114	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 104  ∀wral 2528  ∃!wreu 2530  ∀∃!wralseu 17135

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

This theorem depends on definitions:  df-bi 117  df-ralseu 17137

This theorem is referenced by: (None)