Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj91 Structured version   Unicode version

Theorem bnj91 32156
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 3346 . 2  |-  ( [. Z  /  y ]. ph  <->  [. Z  / 
y ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
3 bnj91.2 . . 3  |-  Z  e. 
_V
43bnj525 32032 . 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 1370    e. wcel 1758   _Vcvv 3070   [.wsbc 3286   (/)c0 3737   ` cfv 5518    predc-bnj14 31978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3072  df-sbc 3287
This theorem is referenced by:  bnj118  32164  bnj125  32167
  Copyright terms: Public domain W3C validator