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Theorem bnj154 28910
Description: Technical lemma for bnj153 28912. 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 3046 . 2  |-  ( [. g  /  f ]. ph'  <->  [. g  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
4 vex 2791 . . 3  |-  g  e. 
_V
5 fveq1 5524 . . . 4  |-  ( f  =  g  ->  (
f `  (/) )  =  ( g `  (/) ) )
65eqeq1d 2291 . . 3  |-  ( f  =  g  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( g `  (/) )  = 
pred ( x ,  A ,  R ) ) )
74, 6sbcie 3025 . 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 1623   [.wsbc 2991   (/)c0 3455   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj153  28912  bnj580  28945  bnj607  28948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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