Theorem bj-equsal2

Description: One direction of equsal . (Contributed by BJ, 30-Sep-2018)

	Ref	Expression
Hypotheses	bj-equsal2.1	$$⊢Ⅎx\phi$$
	bj-equsal2.2	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	bj-equsal2	$$⊢\phi \to \forall x\left(x=y\to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-equsal2.1	$$⊢Ⅎx\phi$$
2		bj-equsal2.2	$$⊢x=y\to \left(\phi \to \psi \right)$$
3	1	bj-equsal1ti	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \phi$$
4	2	a2i	$$⊢\left(x=y\to \phi \right)\to \left(x=y\to \psi \right)$$
5	4	alimi	$$⊢\forall x\left(x=y\to \phi \right)\to \forall x\left(x=y\to \psi \right)$$
6	3 5	sylbir	$$⊢\phi \to \forall x\left(x=y\to \psi \right)$$