Theorem sbcieg

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 12-Oct-2024)

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

Proof

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