Theorem nfalseu 17149

Description: Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 17118. Unlike the set.mm version of this theorem, no disjoint variable condition is needed, because nfeu 2105 here does not require one. (Contributed by David A. Wheeler, 22-Jul-2026.)

Hypotheses

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

Assertion

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

Proof of Theorem nfalseu

Step	Hyp	Ref	Expression
1		df-alseu 17136	. 2 ⊢ (∀∃!𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑))
2		nfalseu.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfalseu.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1625	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 1629	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfeu 2105	. . 3 ⊢ Ⅎ𝑥∃!𝑦𝜑
7	5, 6	nfan 1618	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃!𝑦𝜑)
8	1, 7	nfxfr 1527	1 ⊢ Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400  Ⅎwnf 1513  ∃!weu 2086  ∀∃!walseu 17134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-alseu 17136

This theorem is referenced by: (None)