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Metamath Proof Explorer

Theorem sbco3

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Proof shortened by Wolf Lammen, 18-Sep-2018) (New usage is discouraged.)

Ref Expression
Assertion sbco3 
|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Proof

Step Hyp Ref Expression
1 drsb1 
 |-  ( A. x x = y -> ( [ z / x ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph ) )
2 nfae 
 |-  F/ x A. x x = y
3 sbequ12a 
 |-  ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
4 3 sps 
 |-  ( A. x x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
5 2 4 sbbid 
 |-  ( A. x x = y -> ( [ z / x ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) )
6 1 5 bitr3d 
 |-  ( A. x x = y -> ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) )
7 nfnae 
 |-  F/ y -. A. x x = y
8 nfnae 
 |-  F/ x -. A. x x = y
9 nfsb2 
 |-  ( -. A. x x = y -> F/ x [ y / x ] ph )
10 7 8 9 sbco2d 
 |-  ( -. A. x x = y -> ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph ) )
11 sbco 
 |-  ( [ x / y ] [ y / x ] ph <-> [ x / y ] ph )
12 11 sbbii 
 |-  ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
13 10 12 bitr3di 
 |-  ( -. A. x x = y -> ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) )
14 6 13 pm2.61i 
 |-  ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )