Metamath Proof Explorer

Theorem cbvralsv

Description: Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvralsvw when possible. (Contributed by NM, 20-Nov-2005) (Revised by Andrew Salmon, 11-Jul-2011) (New usage is discouraged.)

Ref Expression
Assertion cbvralsv x A φ y A y x φ


Step Hyp Ref Expression
1 nfv z φ
2 nfs1v x z x φ
3 sbequ12 x = z φ z x φ
4 1 2 3 cbvral x A φ z A z x φ
5 nfv y φ
6 5 nfsb y z x φ
7 nfv z y x φ
8 sbequ z = y z x φ y x φ
9 6 7 8 cbvral z A z x φ y A y x φ
10 4 9 bitri x A φ y A y x φ