Theorem sbceqal

Description: Class version of one implication of equvelv . (Contributed by Andrew Salmon, 28-Jun-2011)

	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		spsbc	$$⊢A\in V\to \left(\forall x\left(x=A\to x=B\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(x=A\to x=B\right)\right)$$
2		sbcimg	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(x=A\to x=B\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=A\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)\right)$$
3		eqid	$$⊢A=A$$
4		eqsbc3	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=A\leftrightarrow A=A\right)$$
5	3 4	mpbiri	$$⊢A\in V\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=A$$
6		pm5.5	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=A\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=A\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)\leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)$$
7	5 6	syl	$$⊢A\in V\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=A\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)\leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)$$
8		eqsbc3	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\leftrightarrow A=B\right)$$
9	2 7 8	3bitrd	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(x=A\to x=B\right)\leftrightarrow A=B\right)$$
10	1 9	sylibd	$$⊢A\in V\to \left(\forall x\left(x=A\to x=B\right)\to A=B\right)$$