Theorem alseud 50891

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 50893 and alseu2d 50894 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 50886	. 2 ⊢ (∀∃!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃!𝑥𝜓))
4	1, 2, 3	sylanbrc 595	1 ⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃!weu 2594  ∀∃!walseu 50884

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-alseu 50886

This theorem is used by: (None)