Theorem sbciedf

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014)

	Ref	Expression
Hypotheses	sbcied.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	sbcied.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
	sbciedf.3	⊢ Ⅎ 𝑥 𝜑
	sbciedf.4	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
Assertion	sbciedf	⊢ ( 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		sbciedf.3	⊢ Ⅎ 𝑥 𝜑
4		sbciedf.4	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
5	2	ex	⊢ ( 𝜑 → ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) )
6	3 5	alrimi	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) )
7		sbciegft	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Ⅎ 𝑥 𝜒 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) ) → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )
8	1 4 6 7	syl3anc	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )