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Metamath Proof Explorer

Theorem csbief

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005) (Revised by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Hypotheses	csbief.1	⊢ 𝐴 ∈ V
	csbief.2	⊢ Ⅎ 𝑥 𝐶
	csbief.3	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
Assertion	csbief	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶

Proof

Step	Hyp	Ref	Expression
1		csbief.1	⊢ 𝐴 ∈ V
2		csbief.2	⊢ Ⅎ 𝑥 𝐶
3		csbief.3	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
4	2	a1i	⊢ ( 𝐴 ∈ V → Ⅎ 𝑥 𝐶 )
5	4 3	csbiegf	⊢ ( 𝐴 ∈ V → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶 )
6	1 5	ax-mp	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶