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Theorem sbc2or 3297
Description: The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for  [ A  /  x ] ph behavior at proper classes, matching the sbc5 3313 (false) and sbc6 3315 (true) conclusions. This is interesting since dfsbcq 3290 and dfsbcq2 3291 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem doesn't tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable  y that  ph or  A may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc2or  |-  ( (
[. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem sbc2or
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3291 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
2 eqeq2 2467 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 704 . . . . 5  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1681 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
5 sb5 2143 . . . 4  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
61, 4, 5vtoclbg 3131 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
76orcd 392 . 2  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) ) )
8 pm5.15 884 . . 3  |-  ( (
[. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  -.  E. x
( x  =  A  /\  ph ) ) )
9 vex 3075 . . . . . . . . . 10  |-  x  e. 
_V
10 eleq1 2524 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
119, 10mpbii 211 . . . . . . . . 9  |-  ( x  =  A  ->  A  e.  _V )
1211adantr 465 . . . . . . . 8  |-  ( ( x  =  A  /\  ph )  ->  A  e.  _V )
1312con3i 135 . . . . . . 7  |-  ( -.  A  e.  _V  ->  -.  ( x  =  A  /\  ph ) )
1413nexdv 1822 . . . . . 6  |-  ( -.  A  e.  _V  ->  -. 
E. x ( x  =  A  /\  ph ) )
1511con3i 135 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  -.  x  =  A )
1615pm2.21d 106 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( x  =  A  ->  ph ) )
1716alrimiv 1686 . . . . . 6  |-  ( -.  A  e.  _V  ->  A. x ( x  =  A  ->  ph ) )
1814, 172thd 240 . . . . 5  |-  ( -.  A  e.  _V  ->  ( -.  E. x ( x  =  A  /\  ph )  <->  A. x ( x  =  A  ->  ph )
) )
1918bibi2d 318 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( [. A  /  x ]. ph  <->  -.  E. x
( x  =  A  /\  ph ) )  <-> 
( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) ) )
2019orbi2d 701 . . 3  |-  ( -.  A  e.  _V  ->  ( ( ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  -.  E. x
( x  =  A  /\  ph ) ) )  <->  ( ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x
( x  =  A  ->  ph ) ) ) ) )
218, 20mpbii 211 . 2  |-  ( -.  A  e.  _V  ->  ( ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x
( x  =  A  ->  ph ) ) ) )
227, 21pm2.61i 164 1  |-  ( (
[. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587   [wsb 1702    e. wcel 1758   _Vcvv 3072   [.wsbc 3288
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-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3074  df-sbc 3289
This theorem is referenced by: (None)
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