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Theorem bnj154 29226
Description: Technical lemma for bnj153 29228. 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
bnj154.1  |-  ( ph1  <->  [. g  /  f ]. ph' )
bnj154.2  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
Assertion
Ref Expression
bnj154  |-  ( ph1  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
Distinct variable groups:    A, f    R, f    f, g    x, f
Allowed substitution hints:    A( x, g)    R( x, g)    ph'( x, f, g)    ph1( x, f, g)

Proof of Theorem bnj154
StepHypRef Expression
1 bnj154.1 . 2  |-  ( ph1  <->  [. g  /  f ]. ph' )
2 bnj154.2 . . 3  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
32sbcbii 3059 . 2  |-  ( [. g  /  f ]. ph'  <->  [. g  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
4 vex 2804 . . 3  |-  g  e. 
_V
5 fveq1 5540 . . . 4  |-  ( f  =  g  ->  (
f `  (/) )  =  ( g `  (/) ) )
65eqeq1d 2304 . . 3  |-  ( f  =  g  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( g `  (/) )  = 
pred ( x ,  A ,  R ) ) )
74, 6sbcie 3038 . 2  |-  ( [. g  /  f ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( g `  (/) )  = 
pred ( x ,  A ,  R ) )
81, 3, 73bitri 262 1  |-  ( ph1  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632   [.wsbc 3004   (/)c0 3468   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj153  29228  bnj580  29261  bnj607  29264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-sbc 3005  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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