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Metamath Proof Explorer

Theorem sbc2ie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008) (Revised by Mario Carneiro, 19-Dec-2013) (Proof shortened by GG, 12-Oct-2024)

	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	2	a1i	⊢ ( 𝑥 = 𝐴 → 𝐵 ∈ V )
5	4 3	sbcied	⊢ ( 𝑥 = 𝐴 → ( [ 𝐵 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
6	1 5	sbcie	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ 𝜓 )