Theorem bj-19.9htbi

Description: Strengthening 19.9ht by replacing its consequent with a biconditional ( 19.9t does have a biconditional consequent). This propagates. (Contributed by BJ, 20-Oct-2019)

	Ref	Expression
Assertion	bj-19.9htbi	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\exists x\phi \leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.9ht	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\exists x\phi \to \phi \right)$$
2		19.8a	$$⊢\phi \to \exists x\phi$$
3	1 2	impbid1	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\exists x\phi \leftrightarrow \phi \right)$$