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Theorem bnj206 31722
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 sbc3an 3249 . 2  |-  ( [. M  /  n ]. ( ph  /\  ps  /\  ch ) 
<->  ( [. M  /  n ]. ph  /\  [. M  /  n ]. ps  /\  [. M  /  n ]. ch ) )
2 bnj206.1 . . . 4  |-  ( ph'  <->  [. M  /  n ]. ph )
32bicomi 202 . . 3  |-  ( [. M  /  n ]. ph  <->  ph' )
4 bnj206.2 . . . 4  |-  ( ps'  <->  [. M  /  n ]. ps )
54bicomi 202 . . 3  |-  ( [. M  /  n ]. ps  <->  ps' )
6 bnj206.3 . . . 4  |-  ( ch'  <->  [. M  /  n ]. ch )
76bicomi 202 . . 3  |-  ( [. M  /  n ]. ch  <->  ch' )
83, 5, 73anbi123i 1176 . 2  |-  ( (
[. M  /  n ]. ph  /\  [. M  /  n ]. ps  /\  [. M  /  n ]. ch )  <->  ( ph'  /\  ps'  /\  ch' ) )
91, 8bitri 249 1  |-  ( [. M  /  n ]. ( ph  /\  ps  /\  ch ) 
<->  ( ph'  /\  ps'  /\  ch' ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 965    e. wcel 1756   _Vcvv 2972   [.wsbc 3186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2974  df-sbc 3187
This theorem is referenced by:  bnj124  31864  bnj207  31874
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