Theorem ralseud 17142

Description: Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 17145 and ralseu2d 17146 taken together. (Contributed by David A. Wheeler, 22-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 17137	. 2 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ∧ ∃!𝑥 ∈ 𝐴 𝜓))
4	1, 2, 3	sylanbrc 421	1 ⊢ (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀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  ax-ia3 108

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

This theorem is referenced by: (None)