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Theorem bnj934 29196
Description: Technical lemma for bnj69 29269. 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
bnj934.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj934.4  |-  ( ph'  <->  [. p  /  n ]. ph )
bnj934.7  |-  ( ph"  <->  [. G  / 
f ]. ph' )
bnj934.50  |-  G  e. 
_V
Assertion
Ref Expression
bnj934  |-  ( (
ph  /\  ( G `  (/) )  =  ( f `  (/) ) )  ->  ph" )
Distinct variable groups:    A, f, n    R, f, n    f, X, n
Allowed substitution hints:    ph( f, n, p)    A( p)    R( p)    G( f, n, p)    X( p)    ph'( f, n, p)    ph"( f, n, p)

Proof of Theorem bnj934
StepHypRef Expression
1 bnj934.1 . . . 4  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 eqtr 2426 . . . 4  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
31, 2sylan2b 473 . . 3  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph )  ->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
4 bnj934.7 . . . . 5  |-  ( ph"  <->  [. G  / 
f ]. ph' )
5 bnj934.4 . . . . . . . 8  |-  ( ph'  <->  [. p  /  n ]. ph )
6 vex 3059 . . . . . . . 8  |-  p  e. 
_V
71, 5, 6bnj523 29148 . . . . . . 7  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)
87, 1bitr4i 252 . . . . . 6  |-  ( ph'  <->  ph )
98sbcbii 3330 . . . . 5  |-  ( [. G  /  f ]. ph'  <->  [. G  / 
f ]. ph )
104, 9bitri 249 . . . 4  |-  ( ph"  <->  [. G  / 
f ]. ph )
11 bnj934.50 . . . 4  |-  G  e. 
_V
121, 10, 11bnj609 29178 . . 3  |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
133, 12sylibr 212 . 2  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph )  ->  ph" )
1413ancoms 451 1  |-  ( (
ph  /\  ( G `  (/) )  =  ( f `  (/) ) )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   _Vcvv 3056   [.wsbc 3274   (/)c0 3735   ` cfv 5523    predc-bnj14 28943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-rex 2757  df-v 3058  df-sbc 3275  df-uni 4189  df-br 4393  df-iota 5487  df-fv 5531
This theorem is referenced by:  bnj929  29197
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