Theorem nfreu 3378

Description: Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfreuw 3376 when possible. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfreu	⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfreu

Step	Hyp	Ref	Expression
1		nftru 1805	. . 3 ⊢ Ⅎ𝑦⊤
2		nfreu.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfreu.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfreud 3374	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1544	1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1538  Ⅎwnf 1784  Ⅎwnfc 2963  ∃!wreu 3142

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-mo 2622  df-eu 2654  df-cleq 2816  df-clel 2895  df-nfc 2965  df-reu 3147

This theorem is referenced by: (None)