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Theorem cgsex2g 2820
Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
Hypotheses
Ref Expression
cgsex2g.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ch )
cgsex2g.2  |-  ( ch 
->  ( ph  <->  ps )
)
Assertion
Ref Expression
cgsex2g  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y ( ch  /\  ph )  <->  ps ) )
Distinct variable groups:    x, y, ps    x, A, y    x, B, y
Allowed substitution hints:    ph( x, y)    ch( x, y)    V( x, y)    W( x, y)

Proof of Theorem cgsex2g
StepHypRef Expression
1 cgsex2g.2 . . . 4  |-  ( ch 
->  ( ph  <->  ps )
)
21biimpa 470 . . 3  |-  ( ( ch  /\  ph )  ->  ps )
32exlimivv 1667 . 2  |-  ( E. x E. y ( ch  /\  ph )  ->  ps )
4 elisset 2798 . . . . . 6  |-  ( A  e.  V  ->  E. x  x  =  A )
5 elisset 2798 . . . . . 6  |-  ( B  e.  W  ->  E. y 
y  =  B )
64, 5anim12i 549 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
7 eeanv 1854 . . . . 5  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
86, 7sylibr 203 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
9 cgsex2g.1 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ch )
1092eximi 1564 . . . 4  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y ch )
118, 10syl 15 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y ch )
121biimprcd 216 . . . . 5  |-  ( ps 
->  ( ch  ->  ph )
)
1312ancld 536 . . . 4  |-  ( ps 
->  ( ch  ->  ( ch  /\  ph ) ) )
14132eximdv 1610 . . 3  |-  ( ps 
->  ( E. x E. y ch  ->  E. x E. y ( ch  /\  ph ) ) )
1511, 14syl5com 26 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ps  ->  E. x E. y ( ch  /\  ph ) ) )
163, 15impbid2 195 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y ( ch  /\  ph )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684
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-ext 2264
This theorem depends on definitions:  df-bi 177  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-v 2790
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