Theorem nfralseu 51114

Description: Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 51083. (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 51101	. 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 1928	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
6	2, 5	nfralw 3309	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝜑 → 𝜓)
7	2, 3	nfreuw 3395	. . 3 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
8	6, 7	nfan 1931	. 2 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑦 ∈ 𝐴 𝜑)
9	1, 8	nfxfr 1885	1 ⊢ Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  Ⅎwnf 1815  Ⅎwnfc 2907  ∀wral 3076  ∃!wreu 3363  ∀∃!wralseu 51099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-mo 2564  df-eu 2594  df-clel 2835  df-nfc 2909  df-ral 3077  df-reu 3366  df-ralseu 51101

This theorem is used by: (None)