# Metamath Proof Explorer

## Theorem nfreu

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

Ref Expression
Hypotheses nfreu.1 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
nfreu.2 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\phi }$
Assertion nfreu ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}\exists !{y}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }$

### Proof

Step Hyp Ref Expression
1 nfreu.1 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
2 nfreu.2 ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\phi }$
3 nftru ${⊢}Ⅎ{y}\phantom{\rule{.4em}{0ex}}\top$
4 1 a1i ${⊢}\top \to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
5 2 a1i ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\phi }$
6 3 4 5 nfreud ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}\exists !{y}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }$
7 6 mptru ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}\exists !{y}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }$