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Theorem cgsex4g 2834
Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
Hypotheses
Ref Expression
cgsex4g.1  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  ->  ch )
cgsex4g.2  |-  ( ch 
->  ( ph  <->  ps )
)
Assertion
Ref Expression
cgsex4g  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( E. x E. y E. z E. w
( ch  /\  ph ) 
<->  ps ) )
Distinct variable groups:    x, y,
z, w, A    x, B, y, z, w    x, C, y, z, w    x, D, y, z, w    ps, x, y, z, w
Allowed substitution hints:    ph( x, y, z, w)    ch( x, y, z, w)    R( x, y, z, w)    S( x, y, z, w)

Proof of Theorem cgsex4g
StepHypRef Expression
1 cgsex4g.2 . . . . 5  |-  ( ch 
->  ( ph  <->  ps )
)
21biimpa 470 . . . 4  |-  ( ( ch  /\  ph )  ->  ps )
32exlimivv 1625 . . 3  |-  ( E. z E. w ( ch  /\  ph )  ->  ps )
43exlimivv 1625 . 2  |-  ( E. x E. y E. z E. w ( ch  /\  ph )  ->  ps )
5 elisset 2811 . . . . . . . 8  |-  ( A  e.  R  ->  E. x  x  =  A )
6 elisset 2811 . . . . . . . 8  |-  ( B  e.  S  ->  E. y 
y  =  B )
75, 6anim12i 549 . . . . . . 7  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
8 eeanv 1866 . . . . . . 7  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
97, 8sylibr 203 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
10 elisset 2811 . . . . . . . 8  |-  ( C  e.  R  ->  E. z 
z  =  C )
11 elisset 2811 . . . . . . . 8  |-  ( D  e.  S  ->  E. w  w  =  D )
1210, 11anim12i 549 . . . . . . 7  |-  ( ( C  e.  R  /\  D  e.  S )  ->  ( E. z  z  =  C  /\  E. w  w  =  D
) )
13 eeanv 1866 . . . . . . 7  |-  ( E. z E. w ( z  =  C  /\  w  =  D )  <->  ( E. z  z  =  C  /\  E. w  w  =  D )
)
1412, 13sylibr 203 . . . . . 6  |-  ( ( C  e.  R  /\  D  e.  S )  ->  E. z E. w
( z  =  C  /\  w  =  D ) )
159, 14anim12i 549 . . . . 5  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( E. x E. y ( x  =  A  /\  y  =  B )  /\  E. z E. w ( z  =  C  /\  w  =  D ) ) )
16 ee4anv 1868 . . . . 5  |-  ( E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  <->  ( E. x E. y ( x  =  A  /\  y  =  B )  /\  E. z E. w ( z  =  C  /\  w  =  D ) ) )
1715, 16sylibr 203 . . . 4  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  ->  E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) ) )
18 cgsex4g.1 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  ->  ch )
19182eximi 1567 . . . . 5  |-  ( E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  ->  E. z E. w ch )
20192eximi 1567 . . . 4  |-  ( E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  ->  E. x E. y E. z E. w ch )
2117, 20syl 15 . . 3  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  ->  E. x E. y E. z E. w ch )
221biimprcd 216 . . . . . 6  |-  ( ps 
->  ( ch  ->  ph )
)
2322ancld 536 . . . . 5  |-  ( ps 
->  ( ch  ->  ( ch  /\  ph ) ) )
24232eximdv 1614 . . . 4  |-  ( ps 
->  ( E. z E. w ch  ->  E. z E. w ( ch  /\  ph ) ) )
25242eximdv 1614 . . 3  |-  ( ps 
->  ( E. x E. y E. z E. w ch  ->  E. x E. y E. z E. w ( ch  /\  ph )
) )
2621, 25syl5com 26 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( ps  ->  E. x E. y E. z E. w ( ch  /\  ph ) ) )
274, 26impbid2 195 1  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( E. x E. y E. z E. w
( ch  /\  ph ) 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  copsex4g  4271  brecop  6767
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-ext 2277
This theorem depends on definitions:  df-bi 177  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-v 2803
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