Theorem sbcbidv

Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014) Drop ax-12 . (Revised by GG, 1-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		sbcbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		eqidd	$$⊢\phi \to A=A$$
3	2 1	sbceqbid	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$