Theorem 19.21h

Description: Theorem 19.21 of Margaris p. 90. The hypothesis can be thought of as " x is not free in ph ". See also 19.21 and 19.21v . (Contributed by NM, 1-Aug-2017) (Proof shortened by Wolf Lammen, 1-Jan-2018)

	Ref	Expression
Hypothesis	19.21h.1	$$⊢\phi \to \forall x\phi$$
Assertion	19.21h	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.21h.1	$$⊢\phi \to \forall x\phi$$
2	1	nf5i	$$⊢Ⅎx\phi$$
3	2	19.21	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)$$