Theorem alseueu 17152

Description: "The 𝜑 is 𝜓 " implies that exactly one thing is both 𝜑 and 𝜓. This is the half of dfalseu2 17151 that drops the universal conjunct; it does not reverse, so ∃!𝑥(𝜑 ∧ 𝜓) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 22-Jul-2026.)

Assertion

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

Proof of Theorem alseueu

Step	Hyp	Ref	Expression
1		dfalseu2 17151	. 2 ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥(𝜑 ∧ 𝜓)))
2	1	simprbi 275	1 ⊢ (∀∃!𝑥(𝜑 → 𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400  ∃!weu 2086  ∀∃!walseu 17134

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-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-eu 2089  df-alseu 17136

This theorem is referenced by: (None)