This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ichv

Description: Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023)

Ref Expression
Assertion ichv 
|- [ x <> y ] ph

Proof

Step Hyp Ref Expression
1 sbv 
 |-  ( [ a / y ] ph <-> ph )
2 1 sbbii 
 |-  ( [ y / x ] [ a / y ] ph <-> [ y / x ] ph )
3 sbv 
 |-  ( [ y / x ] ph <-> ph )
4 2 3 bitri 
 |-  ( [ y / x ] [ a / y ] ph <-> ph )
5 4 sbbii 
 |-  ( [ x / a ] [ y / x ] [ a / y ] ph <-> [ x / a ] ph )
6 sbv 
 |-  ( [ x / a ] ph <-> ph )
7 5 6 bitri 
 |-  ( [ x / a ] [ y / x ] [ a / y ] ph <-> ph )
8 7 gen2 
 |-  A. x A. y ( [ x / a ] [ y / x ] [ a / y ] ph <-> ph )
9 df-ich 
 |-  ( [ x <> y ] ph <-> A. x A. y ( [ x / a ] [ y / x ] [ a / y ] ph <-> ph ) )
10 8 9 mpbir 
 |-  [ x <> y ] ph