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Theorem sbc2or 3012
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 3028 (false) and sbc6 3030 (true) conclusions. This is interesting since dfsbcq 3006 and dfsbcq2 3007 (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 3007 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
2 eqeq2 2305 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 685 . . . . 5  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1616 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
5 sb5 2052 . . . 4  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
61, 4, 5vtoclbg 2857 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
76orcd 381 . 2  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) ) )
8 pm5.15 859 . . 3  |-  ( (
[. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  -.  E. x
( x  =  A  /\  ph ) ) )
9 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
10 eleq1 2356 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
119, 10mpbii 202 . . . . . . . . 9  |-  ( x  =  A  ->  A  e.  _V )
1211adantr 451 . . . . . . . 8  |-  ( ( x  =  A  /\  ph )  ->  A  e.  _V )
1312con3i 127 . . . . . . 7  |-  ( -.  A  e.  _V  ->  -.  ( x  =  A  /\  ph ) )
1413nexdv 1869 . . . . . 6  |-  ( -.  A  e.  _V  ->  -. 
E. x ( x  =  A  /\  ph ) )
1511con3i 127 . . . . . . . 8  |-  ( -.  A  e.  _V  ->  -.  x  =  A )
1615pm2.21d 98 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( x  =  A  ->  ph ) )
1716alrimiv 1621 . . . . . 6  |-  ( -.  A  e.  _V  ->  A. x ( x  =  A  ->  ph ) )
1814, 172thd 231 . . . . 5  |-  ( -.  A  e.  _V  ->  ( -.  E. x ( x  =  A  /\  ph )  <->  A. x ( x  =  A  ->  ph )
) )
1918bibi2d 309 . . . 4  |-  ( -.  A  e.  _V  ->  ( ( [. A  /  x ]. ph  <->  -.  E. x
( x  =  A  /\  ph ) )  <-> 
( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) ) )
2019orbi2d 682 . . 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 202 . 2  |-  ( -.  A  e.  _V  ->  ( ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x
( x  =  A  ->  ph ) ) ) )
227, 21pm2.61i 156 1  |-  ( (
[. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) )  \/  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632   [wsb 1638    e. wcel 1696   _Vcvv 2801   [.wsbc 3004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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