Theorem nfralseu 50613

Description: Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 50582. (Contributed by David A. Wheeler, 21-Jul-2026.)

Hypotheses

Ref	Expression
nfralseu.1	⊢ Ⅎ𝑥𝐴
nfralseu.2	⊢ Ⅎ𝑥𝜑
nfralseu.3	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfralseu	⊢ Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfralseu

Step	Hyp	Ref	Expression
1		df-ralseu 50600	. 2 ⊢ (∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑦 ∈ 𝐴 𝜑))
2		nfralseu.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfralseu.2	. . . . 5 ⊢ Ⅎ𝑥𝜑
4		nfralseu.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfim 1926	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
6	2, 5	nfralw 3312	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝜑 → 𝜓)
7	2, 3	nfreuw 3399	. . 3 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
8	6, 7	nfan 1929	. 2 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑦 ∈ 𝐴 𝜑)
9	1, 8	nfxfr 1883	1 ⊢ Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1813  Ⅎwnfc 2910  ∀wral 3079  ∃!wreu 3367  ∀∃!wralseu 50598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-mo 2567  df-eu 2597  df-clel 2838  df-nfc 2912  df-ral 3080  df-reu 3370  df-ralseu 50600

This theorem is referenced by: (None)