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Theorem bnj91 34301
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 3321 . 2  |-  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
3 bnj91.2 . . 3  |-  Z  e. 
_V
43bnj525 34176 . 2  |-  ( [. Z  /  y ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( x ,  A ,  R ) )
52, 4bitri 249 1  |-  ( [. Z  /  y ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1399    e. wcel 1836   _Vcvv 3047   [.wsbc 3265   (/)c0 3724   ` cfv 5509    predc-bnj14 34122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2378  df-cleq 2384  df-clel 2387  df-v 3049  df-sbc 3266
This theorem is referenced by:  bnj118  34309  bnj125  34312
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