Theorem sbc2ie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	sbc2ie.1	⊢ 𝐴 ∈ V
	sbc2ie.2	⊢ 𝐵 ∈ V
	sbc2ie.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	sbc2ie	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbc2ie.1	⊢ 𝐴 ∈ V
2		sbc2ie.2	⊢ 𝐵 ∈ V
3		sbc2ie.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
4		nfv	⊢ Ⅎ 𝑥 𝜓
5		nfv	⊢ Ⅎ 𝑦 𝜓
6	2	nfth	⊢ Ⅎ 𝑥 𝐵 ∈ V
7	4 5 6 3	sbc2iegf	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
8	1 2 7	mp2an	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ 𝜓 )