Theorem sbcied

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Hypotheses	sbcied.1	$$⊢\phi \to A\in V$$
	sbcied.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	sbcied	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcied.1	$$⊢\phi \to A\in V$$
2		sbcied.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow A\in \left\{x|\psi \right\}$$
4	1 2	elabd3	$$⊢\phi \to \left(A\in \left\{x|\psi \right\}\leftrightarrow \chi \right)$$
5	3 4	bitrid	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \chi \right)$$