Theorem sbcied2

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

	Ref	Expression
Hypotheses	sbcied2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	sbcied2.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	sbcied2.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	sbcied2	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcied2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		sbcied2.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		sbcied2.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) )
4		id	⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 )
5	4 2	sylan9eqr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝑥 = 𝐵 )
6	5 3	syldan	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
7	1 6	sbcied	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) )