Theorem alseud 17339

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 17341 and alseu2d 17342 taken together. (Contributed by David A. Wheeler, 22-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 17334	. 2 ⊢ (∀∃!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃!𝑥𝜓))
4	1, 2, 3	sylanbrc 421	1 ⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1400  ∃!weu 2086  ∀∃!walseu 17332

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-alseu 17334

This theorem is used by: (None)