Theorem nfreud 3433

Description: Deduction version of nfreu 3435. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfrmod.1	⊢ Ⅎ𝑦𝜑
nfrmod.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfrmod.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfreud	⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓)

Proof of Theorem nfreud

Step	Hyp	Ref	Expression
1		df-reu 3381	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfrmod.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvf 2932	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4	3	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
5		nfrmod.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
7	4, 6	nfeld 2917	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
8		nfrmod.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
10	7, 9	nfand 1897	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
11	2, 10	nfeud2 2590	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
12	1, 11	nfxfrd 1854	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538  Ⅎwnf 1783   ∈ wcel 2108  ∃!weu 2568  Ⅎwnfc 2890  ∃!wreu 3378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-eu 2569  df-cleq 2729  df-clel 2816  df-nfc 2892  df-reu 3381

This theorem is referenced by:  nfreu  3435