Theorem ralseud 50605

Description: Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 50608 and ralseu2d 50609 taken together. (Contributed by David A. Wheeler, 21-Jul-2026.)

Hypotheses

Ref	Expression
ralseud.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
ralseud.2	⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
ralseud	⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))

Proof of Theorem ralseud

Step	Hyp	Ref	Expression
1		ralseud.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
2		ralseud.2	. 2 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
3		df-ralseu 50600	. 2 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃!𝑥 ∈ 𝐴 𝜓))
4	1, 2, 3	sylanbrc 594	1 ⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wral 3079  ∃!wreu 3367  ∀∃!wralseu 50598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-ralseu 50600

This theorem is referenced by: (None)