Theorem sbcied

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014)

	Ref	Expression
Hypotheses	sbcied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	sbcied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	sbcied	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		sbcied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3		nfv	⊢ Ⅎ 𝑥 𝜑
4		nfvd	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
5	1 2 3 4	sbciedf	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )