Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj206 Unicode version

Theorem bnj206 28759
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj206.1  |-  ( ph'  <->  [. M  /  n ]. ph )
bnj206.2  |-  ( ps'  <->  [. M  /  n ]. ps )
bnj206.3  |-  ( ch'  <->  [. M  /  n ]. ch )
bnj206.4  |-  M  e. 
_V
Assertion
Ref Expression
bnj206  |-  ( [. M  /  n ]. ( ph  /\  ps  /\  ch ) 
<->  ( ph'  /\  ps'  /\  ch' ) )

Proof of Theorem bnj206
StepHypRef Expression
1 bnj206.4 . 2  |-  M  e. 
_V
2 sbc3ang 3049 . . 3  |-  ( M  e.  _V  ->  ( [. M  /  n ]. ( ph  /\  ps  /\ 
ch )  <->  ( [. M  /  n ]. ph  /\  [. M  /  n ]. ps  /\  [. M  /  n ]. ch ) ) )
3 bnj206.1 . . . . 5  |-  ( ph'  <->  [. M  /  n ]. ph )
43bicomi 193 . . . 4  |-  ( [. M  /  n ]. ph  <->  ph' )
5 bnj206.2 . . . . 5  |-  ( ps'  <->  [. M  /  n ]. ps )
65bicomi 193 . . . 4  |-  ( [. M  /  n ]. ps  <->  ps' )
7 bnj206.3 . . . . 5  |-  ( ch'  <->  [. M  /  n ]. ch )
87bicomi 193 . . . 4  |-  ( [. M  /  n ]. ch  <->  ch' )
94, 6, 83anbi123i 1140 . . 3  |-  ( (
[. M  /  n ]. ph  /\  [. M  /  n ]. ps  /\  [. M  /  n ]. ch )  <->  ( ph'  /\  ps'  /\  ch' ) )
102, 9syl6bb 252 . 2  |-  ( M  e.  _V  ->  ( [. M  /  n ]. ( ph  /\  ps  /\ 
ch )  <->  ( ph'  /\  ps'  /\  ch' ) ) )
111, 10ax-mp 8 1  |-  ( [. M  /  n ]. ( ph  /\  ps  /\  ch ) 
<->  ( ph'  /\  ps'  /\  ch' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  bnj124  28903  bnj207  28913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
  Copyright terms: Public domain W3C validator