Theorem bj-sngleq

Description: Substitution property for sngl . (Contributed by BJ, 6-Oct-2018)

	Ref	Expression
Assertion	bj-sngleq	$$⊢A=B\to snglA=snglB$$

Proof

Step	Hyp	Ref	Expression
1		rexeq	$$⊢A=B\to \left(\exists y\in Ax=\left\{y\right\}\leftrightarrow \exists y\in Bx=\left\{y\right\}\right)$$
2	1	abbidv	$$⊢A=B\to \left\{x|\exists y\in Ax=\left\{y\right\}\right\}=\left\{x|\exists y\in Bx=\left\{y\right\}\right\}$$
3		df-bj-sngl	$$⊢snglA=\left\{x|\exists y\in Ax=\left\{y\right\}\right\}$$
4		df-bj-sngl	$$⊢snglB=\left\{x|\exists y\in Bx=\left\{y\right\}\right\}$$
5	2 3 4	3eqtr4g	$$⊢A=B\to snglA=snglB$$