Theorem 19.21

Description: Theorem 19.21 of Margaris p. 90. The hypothesis can be thought of as " x is not free in ph ". See 19.21v for a version requiring fewer axioms. See also 19.21h . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 24-Sep-2016) df-nf changed. (Revised by Wolf Lammen, 18-Sep-2021)

	Ref	Expression
Hypothesis	19.21.1	$$⊢Ⅎx\phi$$
Assertion	19.21	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.21.1	$$⊢Ⅎx\phi$$
2		19.21t	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$
3	1 2	ax-mp	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$