Metamath Proof Explorer


Theorem sbcnestg

Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sbcnestgw when possible. (Contributed by NM, 27-Nov-2005) (Proof shortened by Mario Carneiro, 11-Nov-2016) (New usage is discouraged.)

Ref Expression
Assertion sbcnestg AV[˙A/x]˙[˙B/y]˙φ[˙A/xB/y]˙φ

Proof

Step Hyp Ref Expression
1 nfv xφ
2 1 ax-gen yxφ
3 sbcnestgf AVyxφ[˙A/x]˙[˙B/y]˙φ[˙A/xB/y]˙φ
4 2 3 mpan2 AV[˙A/x]˙[˙B/y]˙φ[˙A/xB/y]˙φ