Theorem nfreu 3427

Description: Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker nfreuw 3406 when possible. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfrmo.1	⊢ Ⅎ𝑥𝐴
nfrmo.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfreu	⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfreu

Step	Hyp	Ref	Expression
1		nftru 1799	. . 3 ⊢ Ⅎ𝑦⊤
2		nfrmo.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfrmo.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfreud 3425	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1541	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1535  Ⅎwnf 1778  Ⅎwnfc 2879  ∃!wreu 3370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2530  df-eu 2559  df-cleq 2720  df-clel 2806  df-nfc 2881  df-reu 3373

This theorem is referenced by: (None)