Theorem sbcie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004)

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

Proof

Step	Hyp	Ref	Expression
1		sbcie.1	⊢ 𝐴 ∈ V
2		sbcie.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3	2	sbcieg	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
4	1 3	ax-mp	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 )