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Theorem bnj91 29159
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 3208 . 2  |-  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
3 bnj91.2 . . 3  |-  Z  e. 
_V
43bnj525 29033 . 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 1652    e. wcel 1725   _Vcvv 2948   [.wsbc 3153   (/)c0 3620   ` cfv 5446    predc-bnj14 28979
This theorem is referenced by:  bnj118  29167  bnj125  29170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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