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Metamath Proof Explorer

Theorem sbco2d

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses sbco2d.1 
|- F/ x ph
sbco2d.2 
|- F/ z ph
sbco2d.3 
|- ( ph -> F/ z ps )
Assertion sbco2d 
|- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )

Proof

Step Hyp Ref Expression
1 sbco2d.1 
 |-  F/ x ph
2 sbco2d.2 
 |-  F/ z ph
3 sbco2d.3 
 |-  ( ph -> F/ z ps )
4 2 3 nfim1 
 |-  F/ z ( ph -> ps )
5 4 sbco2 
 |-  ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
6 1 sbrim 
 |-  ( [ z / x ] ( ph -> ps ) <-> ( ph -> [ z / x ] ps ) )
7 6 sbbii 
 |-  ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( ph -> [ z / x ] ps ) )
8 2 sbrim 
 |-  ( [ y / z ] ( ph -> [ z / x ] ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
9 7 8 bitri 
 |-  ( [ y / z ] [ z / x ] ( ph -> ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
10 1 sbrim 
 |-  ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
11 5 9 10 3bitr3i 
 |-  ( ( ph -> [ y / z ] [ z / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
12 11 pm5.74ri 
 |-  ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )