Theorem sbceq1dd

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017) (Revised by NM, 30-Jun-2018)

	Ref	Expression
Hypotheses	sbceq1d.1	$$⊢\phi \to A=B$$
	sbceq1dd.2	$$⊢\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi$$
Assertion	sbceq1dd	$$⊢\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\psi$$

Proof

Step	Hyp	Ref	Expression
1		sbceq1d.1	$$⊢\phi \to A=B$$
2		sbceq1dd.2	$$⊢\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi$$
3	1	sbceq1d	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\psi \right)$$
4	2 3	mpbid	$$⊢\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\psi$$