# Metamath Proof Explorer

## Theorem nfrmo

Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfrmow when possible. (Contributed by NM, 16-Jun-2017) (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 nfrmo ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\exists }^{*}{y}\in {A}{\phi }$

### 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 df-rmo ${⊢}{\exists }^{*}{y}\in {A}{\phi }↔{\exists }^{*}{y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
4 nftru ${⊢}Ⅎ{y}\phantom{\rule{.4em}{0ex}}\top$
5 nfcvf ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{y}$
6 1 a1i ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
7 5 6 nfeld ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{y}\in {A}$
8 2 a1i ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\phi }$
9 7 8 nfand ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to Ⅎ{x}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
10 9 adantl ${⊢}\left(\top \wedge ¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\right)\to Ⅎ{x}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
11 4 10 nfmod2 ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\exists }^{*}{y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
12 11 mptru ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\exists }^{*}{y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
13 3 12 nfxfr ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\exists }^{*}{y}\in {A}{\phi }$