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 → ( 𝜑𝜓 ) )
Assertion cbvsbcv ( [ 𝐴 / 𝑥 ] 𝜑[ 𝐴 / 𝑦 ] 𝜓 )

Proof

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