Theorem bj-equsal

Description: Shorter proof of equsal . (Contributed by BJ, 30-Sep-2018) Proof modification is discouraged to avoid using equsal , but "min */exc equsal" is ok. (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-equsal.1	$$⊢Ⅎx\psi$$
2		bj-equsal.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	2	biimpd	$$⊢x=y\to \left(\phi \to \psi \right)$$
4	1 3	bj-equsal1	$$⊢\forall x\left(x=y\to \phi \right)\to \psi$$
5	2	biimprd	$$⊢x=y\to \left(\psi \to \phi \right)$$
6	1 5	bj-equsal2	$$⊢\psi \to \forall x\left(x=y\to \phi \right)$$
7	4 6	impbii	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$