Definition df-alseu 17174

Description: Define "all some one" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and exactly one 𝑥 satisfies 𝜑. (Contributed by David A. Wheeler, 22-Jul-2026.)

Assertion

Ref	Expression
df-alseu	⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))

Detailed syntax breakdown of Definition df-alseu

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		wps	. . 3 wff 𝜓
3		vx	. . 3 setvar 𝑥
4	1, 2, 3	walseu 17172	. 2 wff ∀∃!𝑥(𝜑 → 𝜓)
5	1, 2	wi 4	. . . 4 wff (𝜑 → 𝜓)
6	5, 3	wal 1400	. . 3 wff ∀𝑥(𝜑 → 𝜓)
7	1, 3	weu 2086	. . 3 wff ∃!𝑥𝜑
8	6, 7	wa 104	. 2 wff (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑)
9	4, 8	wb 105	1 wff (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))

This definition is used by:  dfralseu2  17176  alseuals  17177  alseud  17179  alseu1d  17181  alseu2d  17182  alseubii  17185  nfalseu  17187  dfalseu2  17189