Metamath Proof Explorer

Theorem ichbidv

Description: Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024)

Ref Expression
Hypothesis ichbidv.1 
|- ( ph -> ( ps <-> ch ) )
Assertion ichbidv 
|- ( ph -> ( [ x <> y ] ps <-> [ x <> y ] ch ) )

Proof

Step Hyp Ref Expression
1 ichbidv.1 
 |-  ( ph -> ( ps <-> ch ) )
2 1 sbbidv 
 |-  ( ph -> ( [ a / y ] ps <-> [ a / y ] ch ) )
3 2 sbbidv 
 |-  ( ph -> ( [ y / x ] [ a / y ] ps <-> [ y / x ] [ a / y ] ch ) )
4 3 sbbidv 
 |-  ( ph -> ( [ x / a ] [ y / x ] [ a / y ] ps <-> [ x / a ] [ y / x ] [ a / y ] ch ) )
5 4 1 bibi12d 
 |-  ( ph -> ( ( [ x / a ] [ y / x ] [ a / y ] ps <-> ps ) <-> ( [ x / a ] [ y / x ] [ a / y ] ch <-> ch ) ) )
6 5 albidv 
 |-  ( ph -> ( A. y ( [ x / a ] [ y / x ] [ a / y ] ps <-> ps ) <-> A. y ( [ x / a ] [ y / x ] [ a / y ] ch <-> ch ) ) )
7 6 albidv 
 |-  ( ph -> ( A. x A. y ( [ x / a ] [ y / x ] [ a / y ] ps <-> ps ) <-> A. x A. y ( [ x / a ] [ y / x ] [ a / y ] ch <-> ch ) ) )
8 df-ich 
 |-  ( [ x <> y ] ps <-> A. x A. y ( [ x / a ] [ y / x ] [ a / y ] ps <-> ps ) )
9 df-ich 
 |-  ( [ x <> y ] ch <-> A. x A. y ( [ x / a ] [ y / x ] [ a / y ] ch <-> ch ) )
10 7 8 9 3bitr4g 
 |-  ( ph -> ( [ x <> y ] ps <-> [ x <> y ] ch ) )