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Theorem bnj118 29241
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 29090 . . 3  |-  1o  e.  _V
42, 3bnj91 29233 . 2  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
51, 4bitri 242 1  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653   [.wsbc 3162   (/)c0 3629   ` cfv 5455   1oc1o 6718    predc-bnj14 29053
This theorem is referenced by:  bnj151  29249  bnj153  29252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-pw 3802  df-sn 3821  df-suc 4588  df-1o 6725
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