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Theorem isoeq2 6042
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 4216 . . . . 5  |-  ( R  =  T  ->  (
x R y  <->  x T
y ) )
21bibi1d 312 . . . 4  |-  ( R  =  T  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x T y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
322ralbidv 2749 . . 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 686 . 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 5465 . 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 5465 . 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 281 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 178    /\ wa 360    = wceq 1653   A.wral 2707   class class class wbr 4214   -1-1-onto->wf1o 5455   ` cfv 5456    Isom wiso 5457
This theorem is referenced by:  leiso  11710  gtiso  24090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-cleq 2431  df-clel 2434  df-ral 2712  df-br 4215  df-isom 5465
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