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Theorem bnj118 28901
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj118.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj118.2  |-  ( ph'  <->  [. 1o  /  n ]. ph )
Assertion
Ref Expression
bnj118  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
Distinct variable groups:    A, n    R, n    f, n    x, n
Allowed substitution hints:    ph( x, f, n)    A( x, f)    R( x, f)    ph'( x, f, n)

Proof of Theorem bnj118
StepHypRef Expression
1 bnj118.2 . 2  |-  ( ph'  <->  [. 1o  /  n ]. ph )
2 bnj118.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
3 bnj105 28750 . . 3  |-  1o  e.  _V
42, 3bnj91 28893 . 2  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
51, 4bitri 240 1  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   [.wsbc 2991   (/)c0 3455   ` cfv 5255   1oc1o 6472    predc-bnj14 28713
This theorem is referenced by:  bnj151  28909  bnj153  28912
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ne 2448  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-suc 4398  df-1o 6479
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