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Metamath Proof Explorer

Theorem cbvsbcv

Description: Change the bound variable of a class substitution using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvsbcvw when possible. (Contributed by NM, 30-Sep-2008) (Revised by Mario Carneiro, 13-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	cbvsbcv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	Assertion	cbvsbcv	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑦 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvsbcv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	2 3 1	cbvsbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑦 ] 𝜓 )