Theorem nfmod2 2558

Description: Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2372. See nfmodv 2559 for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 2372. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2569. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfmod2.1	⊢ Ⅎ𝑦𝜑
nfmod2.2	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfmod2	⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Proof of Theorem nfmod2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mo 2540	. 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
2		nfv 1917	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfmod2.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfmod2.2	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5		nfeqf1 2379	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
6	5	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
7	4, 6	nfimd 1897	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧))
8	3, 7	nfald2 2445	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧))
9	2, 8	nfexd 2323	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
10	1, 9	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540

This theorem is referenced by:  nfmod  2561  nfeud2  2590  nfrmod  3303  nfrmo  3309  nfdisj  5052