Theorem alseu2d 17144

Description: Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "exactly one" part. (Contributed by David A. Wheeler, 22-Jul-2026.)

Hypothesis

Ref	Expression
alseu2d.1	⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))

Assertion

Ref	Expression
alseu2d	⊢ (𝜑 → ∃!𝑥𝜓)

Proof of Theorem alseu2d

Step	Hyp	Ref	Expression
1		alseu2d.1	. . 3 ⊢ (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))
2		df-alseu 17136	. . 3 ⊢ (∀∃!𝑥(𝜓 → 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∃!𝑥𝜓))
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝜒) ∧ ∃!𝑥𝜓))
4	3	simprd 114	1 ⊢ (𝜑 → ∃!𝑥𝜓)

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

This theorem depends on definitions:  df-bi 117  df-alseu 17136

This theorem is referenced by: (None)