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Theorem bnj609 29225
Description: Technical lemma for bnj852 29229. 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
bnj609.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj609.2  |-  ( ph"  <->  [. G  / 
f ]. ph )
bnj609.3  |-  G  e. 
_V
Assertion
Ref Expression
bnj609  |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
Distinct variable groups:    A, f    R, f    f, X
Allowed substitution hints:    ph( f)    G( f)    ph"( f)

Proof of Theorem bnj609
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 bnj609.2 . 2  |-  ( ph"  <->  [. G  / 
f ]. ph )
2 bnj609.3 . . 3  |-  G  e. 
_V
3 dfsbcq 3155 . . 3  |-  ( e  =  G  ->  ( [. e  /  f ]. ph  <->  [. G  /  f ]. ph ) )
4 fveq1 5719 . . . 4  |-  ( e  =  G  ->  (
e `  (/) )  =  ( G `  (/) ) )
54eqeq1d 2443 . . 3  |-  ( e  =  G  ->  (
( e `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( G `  (/) )  = 
pred ( X ,  A ,  R )
) )
6 bnj609.1 . . . . 5  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
76sbcbii 3208 . . . 4  |-  ( [. e  /  f ]. ph  <->  [. e  / 
f ]. ( f `  (/) )  =  pred ( X ,  A ,  R ) )
8 vex 2951 . . . . 5  |-  e  e. 
_V
9 fveq1 5719 . . . . . 6  |-  ( f  =  e  ->  (
f `  (/) )  =  ( e `  (/) ) )
109eqeq1d 2443 . . . . 5  |-  ( f  =  e  ->  (
( f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( e `  (/) )  = 
pred ( X ,  A ,  R )
) )
118, 10sbcie 3187 . . . 4  |-  ( [. e  /  f ]. (
f `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( e `  (/) )  = 
pred ( X ,  A ,  R )
)
127, 11bitri 241 . . 3  |-  ( [. e  /  f ]. ph  <->  ( e `  (/) )  =  pred ( X ,  A ,  R ) )
132, 3, 5, 12vtoclb 3001 . 2  |-  ( [. G  /  f ]. ph  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
141, 13bitri 241 1  |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2948   [.wsbc 3153   (/)c0 3620   ` cfv 5446    predc-bnj14 28989
This theorem is referenced by:  bnj600  29227  bnj908  29239  bnj934  29243
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-sbc 3154  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454
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