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Theorem spc2ed 28092
Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Hypotheses
Ref Expression
spc2ed.x  |-  F/ x ch
spc2ed.y  |-  F/ y ch
spc2ed.1  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ps  <->  ch )
)
Assertion
Ref Expression
spc2ed  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( ch  ->  E. x E. y ps ) )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y
Allowed substitution hints:    ps( x, y)    ch( x, y)    V( x, y)    W( x, y)

Proof of Theorem spc2ed
StepHypRef Expression
1 elisset 3092 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 3092 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 568 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 2043 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 215 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 nfv 1751 . . . . 5  |-  F/ x ph
7 spc2ed.x . . . . 5  |-  F/ x ch
86, 7nfan 1984 . . . 4  |-  F/ x
( ph  /\  ch )
9 nfv 1751 . . . . . 6  |-  F/ y
ph
10 spc2ed.y . . . . . 6  |-  F/ y ch
119, 10nfan 1984 . . . . 5  |-  F/ y ( ph  /\  ch )
12 anass 653 . . . . . . . 8  |-  ( ( ( ch  /\  ph )  /\  ( x  =  A  /\  y  =  B ) )  <->  ( ch  /\  ( ph  /\  (
x  =  A  /\  y  =  B )
) ) )
13 ancom 451 . . . . . . . . 9  |-  ( ( ch  /\  ph )  <->  (
ph  /\  ch )
)
1413anbi1i 699 . . . . . . . 8  |-  ( ( ( ch  /\  ph )  /\  ( x  =  A  /\  y  =  B ) )  <->  ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B
) ) )
1512, 14bitr3i 254 . . . . . . 7  |-  ( ( ch  /\  ( ph  /\  ( x  =  A  /\  y  =  B ) ) )  <->  ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B
) ) )
16 spc2ed.1 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ps  <->  ch )
)
1716biimparc 489 . . . . . . 7  |-  ( ( ch  /\  ( ph  /\  ( x  =  A  /\  y  =  B ) ) )  ->  ps )
1815, 17sylbir 216 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B ) )  ->  ps )
1918ex 435 . . . . 5  |-  ( (
ph  /\  ch )  ->  ( ( x  =  A  /\  y  =  B )  ->  ps ) )
2011, 19eximd 1933 . . . 4  |-  ( (
ph  /\  ch )  ->  ( E. y ( x  =  A  /\  y  =  B )  ->  E. y ps )
)
218, 20eximd 1933 . . 3  |-  ( (
ph  /\  ch )  ->  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y ps ) )
2221impancom 441 . 2  |-  ( (
ph  /\  E. x E. y ( x  =  A  /\  y  =  B ) )  -> 
( ch  ->  E. x E. y ps ) )
235, 22sylan2 476 1  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( ch  ->  E. x E. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659   F/wnf 1663    e. wcel 1868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3083
This theorem is referenced by:  spc2d  28093  cnvoprab  28302
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