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Theorem bnj591 28943
Description: Technical lemma for bnj852 28953. 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 3046 . 2  |-  ( [. k  /  j ]. th  <->  [. k  /  j ]. ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )
3 vex 2791 . . 3  |-  k  e. 
_V
4 fveq2 5525 . . . . 5  |-  ( j  =  k  ->  (
f `  j )  =  ( f `  k ) )
5 fveq2 5525 . . . . 5  |-  ( j  =  k  ->  (
g `  j )  =  ( g `  k ) )
64, 5eqeq12d 2297 . . . 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 3025 . 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 1623    e. wcel 1684   [.wsbc 2991   ` cfv 5255
This theorem is referenced by:  bnj580  28945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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