Theorem sbc3ie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014) (Revised by Mario Carneiro, 29-Dec-2014)

	Ref	Expression
Hypotheses	sbc3ie.1	⊢ 𝐴 ∈ V
	sbc3ie.2	⊢ 𝐵 ∈ V
	sbc3ie.3	⊢ 𝐶 ∈ V
	sbc3ie.4	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	sbc3ie	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] [ 𝐶 / 𝑧 ] 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbc3ie.1	⊢ 𝐴 ∈ V
2		sbc3ie.2	⊢ 𝐵 ∈ V
3		sbc3ie.3	⊢ 𝐶 ∈ V
4		sbc3ie.4	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
5	3	a1i	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝐶 ∈ V )
6	4	3expa	⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
7	5 6	sbcied	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( [ 𝐶 / 𝑧 ] 𝜑 ↔ 𝜓 ) )
8	1 2 7	sbc2ie	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] [ 𝐶 / 𝑧 ] 𝜑 ↔ 𝜓 )