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Theorem bnj526 29699
Description: Technical lemma for bnj852 29732. 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 3323 . 2  |-  ( [. G  /  f ]. ph  <->  [. G  / 
f ]. ( f `  (/) )  =  pred ( X ,  A ,  R ) )
4 bnj526.3 . . 3  |-  G  e. 
_V
5 fveq1 5864 . . . 4  |-  ( f  =  G  ->  (
f `  (/) )  =  ( G `  (/) ) )
65eqeq1d 2453 . . 3  |-  ( f  =  G  ->  (
( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( G `  (/) )  = 
pred ( X ,  A ,  R )
) )
74, 6sbcie 3302 . 2  |-  ( [. G  /  f ]. (
f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( G `  (/) )  = 
pred ( X ,  A ,  R )
)
81, 3, 73bitri 275 1  |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1444    e. wcel 1887   _Vcvv 3045   [.wsbc 3267   (/)c0 3731   ` cfv 5582    predc-bnj14 29493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rex 2743  df-v 3047  df-sbc 3268  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590
This theorem is referenced by:  bnj607  29727
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