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Theorem sbc2or 3075
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 3091 (false) and sbc6 3093 (true) conclusions. This is interesting since dfsbcq 3069 and dfsbcq2 3070 (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 3070 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
2 eqeq2 2367 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 685 . . . . 5  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1626 . . . 4  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
5 sb5 2105 . . . 4  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
61, 4, 5vtoclbg 2920 . . 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 2867 . . . . . . . . . 10  |-  x  e. 
_V
10 eleq1 2418 . . . . . . . . . 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 1921 . . . . . 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 1631 . . . . . 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 1540   E.wex 1541    = wceq 1642   [wsb 1648    e. wcel 1710   _Vcvv 2864   [.wsbc 3067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-sbc 3068
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