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
|- ( ph -> A. y ph )
Assertion hbsb3
|- ( [ y / x ] ph -> A. x [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 hbsb3.1
 |-  ( ph -> A. y ph )
2 1 sbimi
 |-  ( [ y / x ] ph -> [ y / x ] A. y ph )
3 hbsb2a
 |-  ( [ y / x ] A. y ph -> A. x [ y / x ] ph )
4 2 3 syl
 |-  ( [ y / x ] ph -> A. x [ y / x ] ph )