Metamath Proof Explorer


Theorem sbco

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . See sbcov for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 21-Sep-2018) (New usage is discouraged.)

Ref Expression
Assertion sbco
|- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 sbcom3
 |-  ( [ y / x ] [ x / y ] ph <-> [ y / x ] [ y / y ] ph )
2 sbid
 |-  ( [ y / y ] ph <-> ph )
3 2 sbbii
 |-  ( [ y / x ] [ y / y ] ph <-> [ y / x ] ph )
4 1 3 bitri
 |-  ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )