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Metamath Proof Explorer

Theorem wrecseq123

Description: General equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018) (Proof shortened by Scott Fenton, 17-Nov-2024)

Ref Expression
Assertion wrecseq123 
|- ( ( R = S /\ A = B /\ F = G ) -> wrecs ( R , A , F ) = wrecs ( S , B , G ) )

Proof

Step Hyp Ref Expression
1 coeq1 
 |-  ( F = G -> ( F o. 2nd ) = ( G o. 2nd ) )
2 frecseq123 
 |-  ( ( R = S /\ A = B /\ ( F o. 2nd ) = ( G o. 2nd ) ) -> frecs ( R , A , ( F o. 2nd ) ) = frecs ( S , B , ( G o. 2nd ) ) )
3 1 2 syl3an3 
 |-  ( ( R = S /\ A = B /\ F = G ) -> frecs ( R , A , ( F o. 2nd ) ) = frecs ( S , B , ( G o. 2nd ) ) )
4 df-wrecs 
 |-  wrecs ( R , A , F ) = frecs ( R , A , ( F o. 2nd ) )
5 df-wrecs 
 |-  wrecs ( S , B , G ) = frecs ( S , B , ( G o. 2nd ) )
6 3 4 5 3eqtr4g 
 |-  ( ( R = S /\ A = B /\ F = G ) -> wrecs ( R , A , F ) = wrecs ( S , B , G ) )