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Theorem spc2ed 27036
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 3119 . . . . 5  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 3119 . . . . 5  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 566 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1952 . . . 4  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 212 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
65anim2i 569 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( ph  /\  E. x E. y ( x  =  A  /\  y  =  B ) ) )
7 nfv 1678 . . . . 5  |-  F/ x ph
8 spc2ed.x . . . . 5  |-  F/ x ch
97, 8nfan 1870 . . . 4  |-  F/ x
( ph  /\  ch )
10 nfv 1678 . . . . . 6  |-  F/ y
ph
11 spc2ed.y . . . . . 6  |-  F/ y ch
1210, 11nfan 1870 . . . . 5  |-  F/ y ( ph  /\  ch )
13 spc2ed.1 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ps  <->  ch )
)
1413biimparc 487 . . . . . . 7  |-  ( ( ch  /\  ( ph  /\  ( x  =  A  /\  y  =  B ) ) )  ->  ps )
15 anass 649 . . . . . . . . . 10  |-  ( ( ( ch  /\  ph )  /\  ( x  =  A  /\  y  =  B ) )  <->  ( ch  /\  ( ph  /\  (
x  =  A  /\  y  =  B )
) ) )
1615bicomi 202 . . . . . . . . 9  |-  ( ( ch  /\  ( ph  /\  ( x  =  A  /\  y  =  B ) ) )  <->  ( ( ch  /\  ph )  /\  ( x  =  A  /\  y  =  B
) ) )
17 ancom 450 . . . . . . . . . 10  |-  ( ( ch  /\  ph )  <->  (
ph  /\  ch )
)
1817anbi1i 695 . . . . . . . . 9  |-  ( ( ( ch  /\  ph )  /\  ( x  =  A  /\  y  =  B ) )  <->  ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B
) ) )
1916, 18bitri 249 . . . . . . . 8  |-  ( ( ch  /\  ( ph  /\  ( x  =  A  /\  y  =  B ) ) )  <->  ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B
) ) )
2019imbi1i 325 . . . . . . 7  |-  ( ( ( ch  /\  ( ph  /\  ( x  =  A  /\  y  =  B ) ) )  ->  ps )  <->  ( (
( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B ) )  ->  ps ) )
2114, 20mpbi 208 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B ) )  ->  ps )
2221ex 434 . . . . 5  |-  ( (
ph  /\  ch )  ->  ( ( x  =  A  /\  y  =  B )  ->  ps ) )
2312, 22eximd 1825 . . . 4  |-  ( (
ph  /\  ch )  ->  ( E. y ( x  =  A  /\  y  =  B )  ->  E. y ps )
)
249, 23eximd 1825 . . 3  |-  ( (
ph  /\  ch )  ->  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y ps ) )
2524impancom 440 . 2  |-  ( (
ph  /\  E. x E. y ( x  =  A  /\  y  =  B ) )  -> 
( ch  ->  E. x E. y ps ) )
266, 25syl 16 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 184    /\ wa 369    = wceq 1374   E.wex 1591   F/wnf 1594    e. wcel 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3110
This theorem is referenced by:  spc2d  27037  cnvoprab  27206
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