Theorem nfeud2 2651

Description: Bound-variable hypothesis builder for uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out nfeudw 2652 for a version that replaces the distinctor with a disjoint variable condition, not requiring ax-13 2379. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfeud2.1	⊢ Ⅎ𝑦𝜑
nfeud2.2	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfeud2	⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud2

Step	Hyp	Ref	Expression
1		df-eu 2629	. 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
2		nfeud2.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeud2.2	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	2, 3	nfexd2 2457	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
5	2, 3	nfmod2 2617	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
6	4, 5	nfand 1898	. 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7	1, 6	nfxfrd 1855	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785  ∃*wmo 2596  ∃!weu 2628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629

This theorem is referenced by:  nfeud  2653  nfreud  3325