# Metamath Proof Explorer

## Theorem nfbr

Description: Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 14-Oct-2016)

Ref Expression
Hypotheses nfbr.1 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
nfbr.2 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{R}$
nfbr.3 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{B}$
Assertion nfbr ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{A}{R}{B}$

### Proof

Step Hyp Ref Expression
1 nfbr.1 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
2 nfbr.2 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{R}$
3 nfbr.3 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{B}$
4 1 a1i ${⊢}\top \to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
5 2 a1i ${⊢}\top \to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{R}$
6 3 a1i ${⊢}\top \to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{B}$
7 4 5 6 nfbrd ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{A}{R}{B}$
8 7 mptru ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{A}{R}{B}$