Theorem sbceqal

Description: Class version of one implication of equvelv . (Contributed by Andrew Salmon, 28-Jun-2011) (Proof shortened by SN, 26-Oct-2024)

	Ref	Expression
Assertion	sbceqal	$$⊢A\in V\to \left(\forall x\left(x=A\to x=B\right)\to A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq1	$$⊢x=A\to \left(x=A\leftrightarrow A=A\right)$$
2		eqeq1	$$⊢x=A\to \left(x=B\leftrightarrow A=B\right)$$
3	1 2	imbi12d	$$⊢x=A\to \left(\left(x=A\to x=B\right)\leftrightarrow \left(A=A\to A=B\right)\right)$$
4		eqid	$$⊢A=A$$
5	4	a1bi	$$⊢A=B\leftrightarrow \left(A=A\to A=B\right)$$
6	3 5	bitr4di	$$⊢x=A\to \left(\left(x=A\to x=B\right)\leftrightarrow A=B\right)$$
7	6	spcgv	$$⊢A\in V\to \left(\forall x\left(x=A\to x=B\right)\to A=B\right)$$