Theorem bj-cbv3tb

Description: Closed form of cbv3 . (Contributed by BJ, 2-May-2019)

	Ref	Expression
Assertion	bj-cbv3tb	$$⊢\forall x\forall y\left(x=y\to \left(\phi \to \psi \right)\right)\to \left(\left(\forall yℲx\psi \wedge \forall xℲy\phi \right)\to \left(\forall x\phi \to \forall y\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.9t	$$⊢Ⅎx\psi \to \left(\exists x\psi \leftrightarrow \psi \right)$$
2	1	biimpd	$$⊢Ⅎx\psi \to \left(\exists x\psi \to \psi \right)$$
3	2	alimi	$$⊢\forall yℲx\psi \to \forall y\left(\exists x\psi \to \psi \right)$$
4		nf5r	$$⊢Ⅎy\phi \to \left(\phi \to \forall y\phi \right)$$
5	4	alimi	$$⊢\forall xℲy\phi \to \forall x\left(\phi \to \forall y\phi \right)$$
6		bj-cbv3ta	$$⊢\forall x\forall y\left(x=y\to \left(\phi \to \psi \right)\right)\to \left(\left(\forall y\left(\exists x\psi \to \psi \right)\wedge \forall x\left(\phi \to \forall y\phi \right)\right)\to \left(\forall x\phi \to \forall y\psi \right)\right)$$
7	3 5 6	syl2ani	$$⊢\forall x\forall y\left(x=y\to \left(\phi \to \psi \right)\right)\to \left(\left(\forall yℲx\psi \wedge \forall xℲy\phi \right)\to \left(\forall x\phi \to \forall y\psi \right)\right)$$