Theorem nfalseu 50669

Description: Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 50638. Unlike nfals 50638 it requires 𝑥 and 𝑦 to be disjoint, because the corresponding builder for ∃! is nfeuw 2623, which requires it; the version without that requirement, nfeu 2624, depends on ax-13 2406 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 50656	. 2 ⊢ (∀∃!𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑))
2		nfalseu.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfalseu.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1929	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 2358	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfeuw 2623	. . 3 ⊢ Ⅎ𝑥∃!𝑦𝜑
7	5, 6	nfan 1932	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑)
8	1, 7	nfxfr 1886	1 ⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  Ⅎwnf 1816  ∃!weu 2598  ∀∃!walseu 50654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2569  df-eu 2599  df-alseu 50656

This theorem is used by: (None)