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Theorem bnj591 29282
Description: Technical lemma for bnj852 29292. 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 3216 . 2  |-  ( [. k  /  j ]. th  <->  [. k  /  j ]. ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )
3 vex 2959 . . 3  |-  k  e. 
_V
4 fveq2 5728 . . . . 5  |-  ( j  =  k  ->  (
f `  j )  =  ( f `  k ) )
5 fveq2 5728 . . . . 5  |-  ( j  =  k  ->  (
g `  j )  =  ( g `  k ) )
64, 5eqeq12d 2450 . . . 4  |-  ( j  =  k  ->  (
( f `  j
)  =  ( g `
 j )  <->  ( f `  k )  =  ( g `  k ) ) )
76imbi2d 308 . . 3  |-  ( j  =  k  ->  (
( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) )  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `
 k )  =  ( g `  k
) ) ) )
83, 7sbcie 3195 . 2  |-  ( [. k  /  j ]. (
( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) )  <-> 
( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )
92, 8bitri 241 1  |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  k
)  =  ( g `
 k ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   [.wsbc 3161   ` cfv 5454
This theorem is referenced by:  bnj580  29284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462
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