Metamath Proof Explorer

Theorem nfralw

Description: Bound-variable hypothesis builder for restricted quantification. Version of nfral with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 1-Sep-1999) (Revised by Gino Giotto, 10-Jan-2024)

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

Proof

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