Metamath Proof Explorer


Theorem hbsb2e

Description: Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

Ref Expression
Assertion hbsb2e
|- ( [ y / x ] ph -> A. x [ y / x ] E. y ph )

Proof

Step Hyp Ref Expression
1 sb4e
 |-  ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
2 sb2
 |-  ( A. x ( x = y -> E. y ph ) -> [ y / x ] E. y ph )
3 2 axc4i
 |-  ( A. x ( x = y -> E. y ph ) -> A. x [ y / x ] E. y ph )
4 1 3 syl
 |-  ( [ y / x ] ph -> A. x [ y / x ] E. y ph )