Theorem alseueu

Description: "The ph is ps " implies that exactly one thing is both ph and ps . This is the half of dfalseu2 that drops the universal conjunct; it does not reverse, so E! x ( ph /\ ps ) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Assertion	alseueu	⊢ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) → ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		dfalseu2	⊢ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
2	1	simprbi	⊢ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) → ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) )