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Theorem bnj526 28597
Description: Technical lemma for bnj852 28630. 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.)
Hypotheses
Ref Expression
bnj526.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj526.2  |-  ( ph"  <->  [. G  / 
f ]. ph )
bnj526.3  |-  G  e. 
_V
Assertion
Ref Expression
bnj526  |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
Distinct variable groups:    A, f    f, G    R, f    f, X
Allowed substitution hints:    ph( f)    ph"( f)

Proof of Theorem bnj526
StepHypRef Expression
1 bnj526.2 . 2  |-  ( ph"  <->  [. G  / 
f ]. ph )
2 bnj526.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
32sbcbii 3159 . 2  |-  ( [. G  /  f ]. ph  <->  [. G  / 
f ]. ( f `  (/) )  =  pred ( X ,  A ,  R ) )
4 bnj526.3 . . 3  |-  G  e. 
_V
5 fveq1 5667 . . . 4  |-  ( f  =  G  ->  (
f `  (/) )  =  ( G `  (/) ) )
65eqeq1d 2395 . . 3  |-  ( f  =  G  ->  (
( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( G `  (/) )  = 
pred ( X ,  A ,  R )
) )
74, 6sbcie 3138 . 2  |-  ( [. G  /  f ]. (
f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( G `  (/) )  = 
pred ( X ,  A ,  R )
)
81, 3, 73bitri 263 1  |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2899   [.wsbc 3104   (/)c0 3571   ` cfv 5394    predc-bnj14 28390
This theorem is referenced by:  bnj607  28625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-v 2901  df-sbc 3105  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402
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