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Theorem isoeq2 5385
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq2  R  T  H  Isom  R ,  S  ,  H  Isom  T ,  S  ,

Proof of Theorem isoeq2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3757 . . . . 5  R  T  R  T
21bibi1d 222 . . . 4  R  T  R  H `
 S H `  T  H `  S H `
322ralbidv 2342 . . 3  R  T  R  H `
 S H `  T  H `  S H `
43anbi2d 437 . 2  R  T  H : -1-1-onto->  R  H `  S H `
 H :
-1-1-onto->  T  H `  S H `
5 df-isom 4854 . 2  H 
Isom  R ,  S  ,  H : -1-1-onto->  R  H `
 S H `
6 df-isom 4854 . 2  H 
Isom  T ,  S  ,  H : -1-1-onto->  T  H `
 S H `
74, 5, 63bitr4g 212 1  R  T  H  Isom  R ,  S  ,  H  Isom  T ,  S  ,
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wral 2300   class class class wbr 3755   -1-1-onto->wf1o 4844   ` cfv 4845    Isom wiso 4846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-ral 2305  df-br 3756  df-isom 4854
This theorem is referenced by: (None)
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