Theorem alseud 50604

Description: Introduction rule: "all some one" holds if the "for all" part holds and the antecedent has exactly one witness. This is the converse of alseu1d 50606 and alseu2d 50607 taken together. (Contributed by David A. Wheeler, 21-Jul-2026.)

Hypotheses

Ref	Expression
alseud.1	⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
alseud.2	⊢ (𝜑 → ∃!𝑥𝜓)

Assertion

Ref	Expression
alseud	⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))

Proof of Theorem alseud

Step	Hyp	Ref	Expression
1		alseud.1	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
2		alseud.2	. 2 ⊢ (𝜑 → ∃!𝑥𝜓)
3		df-alseu 50599	. 2 ⊢ (∀∃!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃!𝑥𝜓))
4	1, 2, 3	sylanbrc 594	1 ⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1568  ∃!weu 2596  ∀∃!walseu 50597

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-alseu 50599

This theorem is referenced by: (None)