Theorem alseuals

Description: "All some one" implies "all some": requiring exactly one witness is stronger than requiring at least one. Any consequence of an allsome statement is therefore a consequence of the corresponding "all some one" statement, which is how alseu-no-surprise is proved. (Contributed by David A. Wheeler, 21-Jul-2026)

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

Proof

Step	Hyp	Ref	Expression
1		euex	⊢ ( ∃! 𝑥 𝜑 → ∃ 𝑥 𝜑 )
2	1	anim2i	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃! 𝑥 𝜑 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 𝜑 ) )
3		df-alseu	⊢ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃! 𝑥 𝜑 ) )
4		df-als	⊢ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 𝜑 ) )
5	2 3 4	3imtr4i	⊢ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) → ∀∃ 𝑥 ( 𝜑 → 𝜓 ) )