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Metamath Proof Explorer

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 GG, 12-Oct-2024)

	Ref	Expression
Hypotheses	sbcied.1	$⊢ φ \to A \in V$
	sbcied.2	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
Assertion	sbcied	$⊢ φ \to (\underset{˙}{[} A / x \underset{˙}{]} ψ \leftrightarrow χ)$

Proof

Step	Hyp	Ref	Expression
1		sbcied.1	$⊢ φ \to A \in V$
2		sbcied.2	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
3		df-sbc	$⊢ \underset{˙}{[} A / x \underset{˙}{]} ψ \leftrightarrow A \in \{x \| ψ\}$
4	1 2	elabd3	$⊢ φ \to (A \in \{x \| ψ\} \leftrightarrow χ)$
5	3 4	bitrid	$⊢ φ \to (\underset{˙}{[} A / x \underset{˙}{]} ψ \leftrightarrow χ)$