Metamath Proof Explorer


Theorem 2sbiev

Description: Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See 2sbievw for a version with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023) (New usage is discouraged.)

Ref Expression
Hypothesis 2sbiev.1
|- ( ( x = t /\ y = u ) -> ( ph <-> ps ) )
Assertion 2sbiev
|- ( [ t / x ] [ u / y ] ph <-> ps )

Proof

Step Hyp Ref Expression
1 2sbiev.1
 |-  ( ( x = t /\ y = u ) -> ( ph <-> ps ) )
2 nfv
 |-  F/ x ps
3 1 sbiedv
 |-  ( x = t -> ( [ u / y ] ph <-> ps ) )
4 2 3 sbie
 |-  ( [ t / x ] [ u / y ] ph <-> ps )