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Theorem sbc2or 2999
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 3015 (false) and sbc6 3017 (true) conclusions. This is interesting since dfsbcq 2993 and dfsbcq2 2994 (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 2994 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
2 eqeq2 2292 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 685 . . . . 5  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1612 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
5 sb5 2039 . . . 4  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
61, 4, 5vtoclbg 2844 . . 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 2791 . . . . . . . . . 10  |-  x  e. 
_V
10 eleq1 2343 . . . . . . . . . 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 1857 . . . . . 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 1617 . . . . . 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 1527   E.wex 1528    = wceq 1623   [wsb 1629    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
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-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
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