Metamath Proof Explorer


Theorem hbsb3

Description: If y is not free in ph , x is not free in [ y / x ] ph . Usage of this theorem is discouraged because it depends on ax-13 . Check out bj-hbsb3v for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993) (New usage is discouraged.)

Ref Expression
Hypothesis hbsb3.1 φyφ
Assertion hbsb3 yxφxyxφ

Proof

Step Hyp Ref Expression
1 hbsb3.1 φyφ
2 1 sbimi yxφyxyφ
3 hbsb2a yxyφxyxφ
4 2 3 syl yxφxyxφ