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Theorem bnj591 29259
Description: Technical lemma for bnj852 29269. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj591.1  |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
Assertion
Ref Expression
bnj591  |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  k
)  =  ( g `
 k ) ) )
Distinct variable groups:    D, j    ch, j    j, ch'    f, j    g, j    j, k    j, n
Allowed substitution hints:    ch( f, g, k, n)    th( f,
g, j, k, n)    D( f, g, k, n)    ch'( f, g, k, n)

Proof of Theorem bnj591
StepHypRef Expression
1 bnj591.1 . . 3  |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
21sbcbii 3059 . 2  |-  ( [. k  /  j ]. th  <->  [. k  /  j ]. ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )
3 vex 2804 . . 3  |-  k  e. 
_V
4 fveq2 5541 . . . . 5  |-  ( j  =  k  ->  (
f `  j )  =  ( f `  k ) )
5 fveq2 5541 . . . . 5  |-  ( j  =  k  ->  (
g `  j )  =  ( g `  k ) )
64, 5eqeq12d 2310 . . . 4  |-  ( j  =  k  ->  (
( f `  j
)  =  ( g `
 j )  <->  ( f `  k )  =  ( g `  k ) ) )
76imbi2d 307 . . 3  |-  ( j  =  k  ->  (
( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) )  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `
 k )  =  ( g `  k
) ) ) )
83, 7sbcie 3038 . 2  |-  ( [. k  /  j ]. (
( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )
92, 8bitri 240 1  |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  k
)  =  ( g `
 k ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   [.wsbc 3004   ` cfv 5271
This theorem is referenced by:  bnj580  29261
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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