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Theorem isoeq2 5833
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq2  |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  T ,  S  ( A ,  B ) ) )

Proof of Theorem isoeq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4041 . . . . 5  |-  ( R  =  T  ->  (
x R y  <->  x T
y ) )
21bibi1d 310 . . . 4  |-  ( R  =  T  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x T y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
322ralbidv 2598 . . 3  |-  ( R  =  T  ->  ( A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <->  A. x  e.  A  A. y  e.  A  ( x T y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
43anbi2d 684 . 2  |-  ( R  =  T  ->  (
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) )  <->  ( H : A
-1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x T y  <->  ( H `  x ) S ( H `  y ) ) ) ) )
5 df-isom 5280 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
6 df-isom 5280 . 2  |-  ( H 
Isom  T ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x T y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
74, 5, 63bitr4g 279 1  |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  T ,  S  ( A ,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632   A.wral 2556   class class class wbr 4039   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
This theorem is referenced by:  leiso  11413  gtiso  23256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-cleq 2289  df-clel 2292  df-ral 2561  df-br 4040  df-isom 5280
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