# Metamath Proof Explorer

## Theorem 19.21v

Description: Version of 19.21 with a disjoint variable condition, requiring fewer axioms.

Notational convention: We sometimes suffix with "v" the label of a theorem using a distinct variable ("dv") condition instead of a non-freeness hypothesis such as F/ x ph . Conversely, we sometimes suffix with "f" the label of a theorem introducing such a non-freeness hypothesis ("f" stands for "not free in", see df-nf ) instead of a disjoint variable condition. For instance, 19.21v versus 19.21 and vtoclf versus vtocl . Note that "not free in" is less restrictive than "does not occur in". Note that the version with a disjoint variable condition is easily proved from the version with the corresponding non-freeness hypothesis, by using nfv . However, the dv version can often be proved from fewer axioms. (Contributed by NM, 21-Jun-1993) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020) (Proof shortened by Wolf Lammen, 12-Jul-2020)

Ref Expression
Assertion 19.21v ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)↔\left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)$

### Proof

Step Hyp Ref Expression
1 stdpc5v ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)$
2 ax5e ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to {\phi }$
3 2 imim1i ${⊢}\left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)\to \left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)$
4 19.38 ${⊢}\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)$
5 3 4 syl ${⊢}\left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)$
6 1 5 impbii ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\psi }\right)↔\left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}{\psi }\right)$