Theorem alsanmo 17059

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 17036	. . 3 ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
2	1	anbi1i 462	. 2 ⊢ ((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) ∧ ∃*𝑥𝜑))
3		anass 405	. 2 ⊢ (((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)))
4		eu5 2134	. . . 4 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
5	4	bicomi 132	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ ∃!𝑥𝜑)
6	5	anbi2i 461	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
7	2, 3, 6	3bitri 206	1 ⊢ ((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400  ∃wex 1545  ∃!weu 2086  ∃*wmo 2087  ∀∃wals 17034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-als 17036

This theorem is referenced by: (None)