Theorem nfreud 3410

Description: Deduction version of nfreu 3412. Usage of this theorem is discouraged because it depends on ax-13 2402. (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 3367	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfrmod.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvf 2949	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
4	3	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦)
5		nfrmod.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	5	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴)
7	4, 6	nfeld 2934	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴)
8		nfrmod.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
9	8	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
10	7, 9	nfand 1916	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
11	2, 10	nfeud2 2616	. 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
12	1, 11	nfxfrd 1873	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557  Ⅎwnf 1802   ∈ wcel 2141  ∃!weu 2594  Ⅎwnfc 2908  ∃!wreu 3364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-mo 2565  df-eu 2595  df-cleq 2753  df-clel 2836  df-nfc 2910  df-reu 3367

This theorem is referenced by:  nfreu  3412