Theorem ralseu2d 50666

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, 21-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 50657	. . 3 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃!𝑥 ∈ 𝐴 𝜓))
3	1, 2	sylib 221	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃!𝑥 ∈ 𝐴 𝜓))
4	3	simprd 501	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wral 3081  ∃!wreu 3369  ∀∃!wralseu 50655

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-ralseu 50657

This theorem is used by: (None)