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Theorem bnj154 34356
Description: Technical lemma for bnj153 34358. 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 3380 . 2  |-  ( [. g  /  f ]. ph'  <->  [. g  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
4 vex 3109 . . 3  |-  g  e. 
_V
5 fveq1 5847 . . . 4  |-  ( f  =  g  ->  (
f `  (/) )  =  ( g `  (/) ) )
65eqeq1d 2456 . . 3  |-  ( f  =  g  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( g `  (/) )  = 
pred ( x ,  A ,  R ) ) )
74, 6sbcie 3359 . 2  |-  ( [. g  /  f ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( g `  (/) )  = 
pred ( x ,  A ,  R ) )
81, 3, 73bitri 271 1  |-  ( ph1  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398   [.wsbc 3324   (/)c0 3783   ` cfv 5570    predc-bnj14 34160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-v 3108  df-sbc 3325  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578
This theorem is referenced by:  bnj153  34358  bnj580  34391  bnj607  34394
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