Metamath Proof Explorer

Theorem nfrmow

Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 16-Jun-2017) (Revised by Gino Giotto, 10-Jan-2024)

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

Proof

Step Hyp Ref Expression
1 nfreuw.1 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
2 nfreuw.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 nfcvd ${⊢}\top \to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{y}$
6 1 a1i ${⊢}\top \to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
7 5 6 nfeld ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{y}\in {A}$
8 2 a1i ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\phi }$
9 7 8 nfand ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
10 4 9 nfmodv ${⊢}\top \to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\exists }^{*}{y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
11 10 mptru ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\exists }^{*}{y}\phantom{\rule{.4em}{0ex}}\left({y}\in {A}\wedge {\phi }\right)$
12 3 11 nfxfr ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{\exists }^{*}{y}\in {A}{\phi }$