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Theorem bnj91 28570
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 ) )
21sbcbii 3159 . 2  |-  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
3 bnj91.2 . . 3  |-  Z  e. 
_V
43bnj525 28444 . 2  |-  ( [. Z  /  y ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( x ,  A ,  R ) )
52, 4bitri 241 1  |-  ( [. Z  /  y ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2899   [.wsbc 3104   (/)c0 3571   ` cfv 5394    predc-bnj14 28390
This theorem is referenced by:  bnj118  28578  bnj125  28581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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