Theorem alsanmo 50537

Description: An "all some" statement conjoined with the claim that at most one 𝑥 satisfies its antecedent is equivalent to the universal part conjoined with the claim that exactly one 𝑥 satisfies the antecedent. The "all some" quantifier supplies the existence of such an 𝑥 and ∃*𝑥𝜑 supplies the at-most-one part, so together they yield ∃!𝑥𝜑. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)

Assertion

Ref	Expression
alsanmo	⊢ ((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))

Proof of Theorem alsanmo

Step	Hyp	Ref	Expression
1		df-als 50515	. . 3 ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
2	1	anbi1i 635	. 2 ⊢ ((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) ∧ ∃*𝑥𝜑))
3		anass 473	. 2 ⊢ (((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)))
4		df-eu 2604	. . . 4 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
5	4	bicomi 227	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ ∃!𝑥𝜑)
6	5	anbi2i 634	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
7	2, 3, 6	3bitri 300	1 ⊢ ((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566  ∃wex 1807  ∃*wmo 2572  ∃!weu 2603  ∀∃wals 50513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-eu 2604  df-als 50515

This theorem is referenced by: (None)