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	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		stdpc5v	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\phi \to \forall x\psi \right)$$
2		ax5e	$$⊢\exists x\phi \to \phi$$
3	2	imim1i	$$⊢\left(\phi \to \forall x\psi \right)\to \left(\exists x\phi \to \forall x\psi \right)$$
4		19.38	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
5	3 4	syl	$$⊢\left(\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
6	1 5	impbii	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$