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Theorem bnj91 29209
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj91.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj91.2  |-  Z  e. 
_V
Assertion
Ref Expression
bnj91  |-  ( [. Z  /  y ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
Distinct variable groups:    y, A    y, R    y, f    x, y
Allowed substitution hints:    ph( x, y, f)    A( x, f)    R( x, f)    Z( x, y, f)

Proof of Theorem bnj91
StepHypRef Expression
1 bnj91.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
2 bnj91.2 . . 3  |-  Z  e. 
_V
31, 2bnj524 29082 . 2  |-  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
42bnj525 29083 . 2  |-  ( [. Z  /  y ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( x ,  A ,  R ) )
53, 4bitri 240 1  |-  ( [. Z  /  y ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   (/)c0 3468   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj118  29217  bnj125  29220
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-v 2803  df-sbc 3005
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