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Theorem bnj118 29217
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 29066 . . 3  |-  1o  e.  _V
42, 3bnj91 29209 . 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 1632   [.wsbc 3004   (/)c0 3468   ` cfv 5271   1oc1o 6488    predc-bnj14 29029
This theorem is referenced by:  bnj151  29225  bnj153  29228
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ne 2461  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-suc 4414  df-1o 6495
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