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Theorem bnj206 29098
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 3219 . . 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 194 . . . 4  |-  ( [. M  /  n ]. ph  <->  ph' )
5 bnj206.2 . . . . 5  |-  ( ps'  <->  [. M  /  n ]. ps )
65bicomi 194 . . . 4  |-  ( [. M  /  n ]. ps  <->  ps' )
7 bnj206.3 . . . . 5  |-  ( ch'  <->  [. M  /  n ]. ch )
87bicomi 194 . . . 4  |-  ( [. M  /  n ]. ch  <->  ch' )
94, 6, 83anbi123i 1142 . . 3  |-  ( (
[. M  /  n ]. ph  /\  [. M  /  n ]. ps  /\  [. M  /  n ]. ch )  <->  ( ph'  /\  ps'  /\  ch' ) )
102, 9syl6bb 253 . 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 177    /\ w3a 936    e. wcel 1725   _Vcvv 2956   [.wsbc 3161
This theorem is referenced by:  bnj124  29242  bnj207  29252
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 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162
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