# Metamath Proof Explorer

## Theorem nfrex

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 30-Dec-2019) Add disjoint variable condition to avoid ax-13 . See nfrexg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

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

### Proof

Step Hyp Ref Expression
1 nfrex.1 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
2 nfrex.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 nfrexd ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}\exists {y}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }$
7 6 mptru ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}\exists {y}\in {A}\phantom{\rule{.4em}{0ex}}{\phi }$