Theorem eleq2w2ALT

Description: Alternate proof of eleq2w2 and special instance of eleq2 . (Contributed by BJ, 22-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	eleq2w2ALT	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfcleq	$$⊢A=B\leftrightarrow \forall y\left(y\in A\leftrightarrow y\in B\right)$$
2	1	biimpi	$$⊢A=B\to \forall y\left(y\in A\leftrightarrow y\in B\right)$$
3		eleq1w	$$⊢y=x\to \left(y\in A\leftrightarrow x\in A\right)$$
4		eleq1w	$$⊢y=x\to \left(y\in B\leftrightarrow x\in B\right)$$
5	3 4	bibi12d	$$⊢y=x\to \left(\left(y\in A\leftrightarrow y\in B\right)\leftrightarrow \left(x\in A\leftrightarrow x\in B\right)\right)$$
6	5	spvv	$$⊢\forall y\left(y\in A\leftrightarrow y\in B\right)\to \left(x\in A\leftrightarrow x\in B\right)$$
7	2 6	syl	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$