Theorem csbied

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014) (Revised by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Hypotheses	csbied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	csbied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
Assertion	csbied	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		csbied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		csbied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐵 = 𝐶 )
3		nfv	⊢ Ⅎ 𝑥 𝜑
4		nfcvd	⊢ ( 𝜑 → Ⅎ 𝑥 𝐶 )
5	3 4 1 2	csbiedf	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )