Theorem nfreu 3396

Description: Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker nfreuw 3378 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 1805	. . 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 3394	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1548	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1542  Ⅎwnf 1784  Ⅎwnfc 2881  ∃!wreu 3346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-mo 2537  df-eu 2567  df-cleq 2726  df-clel 2809  df-nfc 2883  df-reu 3349

This theorem is referenced by: (None)