Theorem nfreu 3415

Description: Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2405. Use the weaker nfreuw 3399 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 1826	. . 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 3413	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1569	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1563  Ⅎwnf 1805  Ⅎwnfc 2911  ∃!wreu 3367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-13 2405  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-mo 2568  df-eu 2598  df-cleq 2756  df-clel 2839  df-nfc 2913  df-reu 3370

This theorem is referenced by: (None)