Theorem nfalseu 51113

Description: Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 51082. Unlike nfals 51082 it requires 𝑥 and 𝑦 to be disjoint, because the corresponding builder for ∃! is nfeuw 2618, which requires it; the version without that requirement, nfeu 2619, depends on ax-13 2401 and its use is discouraged. (Contributed by David A. Wheeler, 21-Jul-2026.)

Hypotheses

Ref	Expression
nfalseu.1	⊢ Ⅎ𝑥𝜑
nfalseu.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfalseu	⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfalseu

Step	Hyp	Ref	Expression
1		df-alseu 51100	. 2 ⊢ (∀∃!𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑))
2		nfalseu.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfalseu.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1928	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 2353	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfeuw 2618	. . 3 ⊢ Ⅎ𝑥∃!𝑦𝜑
7	5, 6	nfan 1931	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑)
8	1, 7	nfxfr 1885	1 ⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  Ⅎwnf 1815  ∃!weu 2593  ∀∃!walseu 51098

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-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-alseu 51100

This theorem is used by: (None)