Theorem bj-equsvt

Description: A variant of equsv . (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Assertion	bj-equsvt	$$⊢Ⅎ\text{'}x\phi \to \left(\forall x\left(x=y\to \phi \right)\leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-19.23t	$$⊢Ⅎ\text{'}x\phi \to \left(\forall x\left(x=y\to \phi \right)\leftrightarrow \left(\exists xx=y\to \phi \right)\right)$$
2		ax6ev	$$⊢\exists xx=y$$
3	2	a1bi	$$⊢\phi \leftrightarrow \left(\exists xx=y\to \phi \right)$$
4	1 3	bitr4di	$$⊢Ⅎ\text{'}x\phi \to \left(\forall x\left(x=y\to \phi \right)\leftrightarrow \phi \right)$$