Theorem bj-biexal1

Description: A general FOL biconditional that generalizes 19.9ht among others. For this and the following theorems, see also 19.35 , 19.21 , 19.23 . When ph is substituted for ps , both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019)

	Ref	Expression
Assertion	bj-biexal1	$$⊢\forall x\left(\phi \to \forall x\psi \right)\leftrightarrow \left(\exists x\phi \to \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall x\psi$$
2	1	19.23	$$⊢\forall x\left(\phi \to \forall x\psi \right)\leftrightarrow \left(\exists x\phi \to \forall x\psi \right)$$