Theorem bj-19.21t0

Description: Proof of 19.21t from stdpc5t . (Contributed by BJ, 15-Sep-2018) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-19.21t0	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		stdpc5t	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\phi \to \forall x\psi \right)\right)$$
2		19.9t	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$
3	2	imbi1d	$$⊢Ⅎx\phi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$
4		19.38	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
5	3 4	biimtrrdi	$$⊢Ⅎx\phi \to \left(\left(\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)\right)$$
6	1 5	impbid	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$