Theorem nfalseu 50612

Description: Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 50581. Unlike nfals 50581 it requires 𝑥 and 𝑦 to be disjoint, because the corresponding builder for ∃! is nfeuw 2621, which requires it; the version without that requirement, nfeu 2622, depends on ax-13 2404 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 50599	. 2 ⊢ (∀∃!𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑))
2		nfalseu.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfalseu.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1926	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 2356	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfeuw 2621	. . 3 ⊢ Ⅎ𝑥∃!𝑦𝜑
7	5, 6	nfan 1929	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑)
8	1, 7	nfxfr 1883	1 ⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568  Ⅎwnf 1813  ∃!weu 2596  ∀∃!walseu 50597

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-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-alseu 50599

This theorem is referenced by: (None)