Theorem sbceq1d

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

	Ref	Expression
Hypothesis	sbceq1d.1	$$⊢\phi \to A=B$$
Assertion	sbceq1d	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\psi \right)$$

Proof

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