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Metamath Proof Explorer

Theorem vtocl4ga

Description: Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019) (Proof shortened by Wolf Lammen, 31-May-2025)

Ref Expression
Hypotheses vtocl4ga.1 
|- ( x = A -> ( ph <-> ps ) )
vtocl4ga.2 
|- ( y = B -> ( ps <-> ch ) )
vtocl4ga.3 
|- ( z = C -> ( ch <-> rh ) )
vtocl4ga.4 
|- ( w = D -> ( rh <-> th ) )
vtocl4ga.5 
|- ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph )
Assertion vtocl4ga 
|- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )

Proof

Step Hyp Ref Expression
1 vtocl4ga.1 
 |-  ( x = A -> ( ph <-> ps ) )
2 vtocl4ga.2 
 |-  ( y = B -> ( ps <-> ch ) )
3 vtocl4ga.3 
 |-  ( z = C -> ( ch <-> rh ) )
4 vtocl4ga.4 
 |-  ( w = D -> ( rh <-> th ) )
5 vtocl4ga.5 
 |-  ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph )
6 3 imbi2d 
 |-  ( z = C -> ( ( ( A e. Q /\ B e. R ) -> ch ) <-> ( ( A e. Q /\ B e. R ) -> rh ) ) )
7 4 imbi2d 
 |-  ( w = D -> ( ( ( A e. Q /\ B e. R ) -> rh ) <-> ( ( A e. Q /\ B e. R ) -> th ) ) )
8 1 imbi2d 
 |-  ( x = A -> ( ( ( z e. S /\ w e. T ) -> ph ) <-> ( ( z e. S /\ w e. T ) -> ps ) ) )
9 2 imbi2d 
 |-  ( y = B -> ( ( ( z e. S /\ w e. T ) -> ps ) <-> ( ( z e. S /\ w e. T ) -> ch ) ) )
10 5 ex 
 |-  ( ( x e. Q /\ y e. R ) -> ( ( z e. S /\ w e. T ) -> ph ) )
11 8 9 10 vtocl2ga 
 |-  ( ( A e. Q /\ B e. R ) -> ( ( z e. S /\ w e. T ) -> ch ) )
12 11 com12 
 |-  ( ( z e. S /\ w e. T ) -> ( ( A e. Q /\ B e. R ) -> ch ) )
13 6 7 12 vtocl2ga 
 |-  ( ( C e. S /\ D e. T ) -> ( ( A e. Q /\ B e. R ) -> th ) )
14 13 impcom 
 |-  ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )