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Theorem spc2ed 5882
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 2585 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2585 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 325 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1823 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 141 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 nfv 1437 . . . . 5  |-  F/ x ph
7 spc2ed.x . . . . 5  |-  F/ x ch
86, 7nfan 1473 . . . 4  |-  F/ x
( ph  /\  ch )
9 nfv 1437 . . . . . 6  |-  F/ y
ph
10 spc2ed.y . . . . . 6  |-  F/ y ch
119, 10nfan 1473 . . . . 5  |-  F/ y ( ph  /\  ch )
12 anass 387 . . . . . . . 8  |-  ( ( ( ch  /\  ph )  /\  ( x  =  A  /\  y  =  B ) )  <->  ( ch  /\  ( ph  /\  (
x  =  A  /\  y  =  B )
) ) )
13 ancom 257 . . . . . . . . 9  |-  ( ( ch  /\  ph )  <->  (
ph  /\  ch )
)
1413anbi1i 439 . . . . . . . 8  |-  ( ( ( ch  /\  ph )  /\  ( x  =  A  /\  y  =  B ) )  <->  ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B
) ) )
1512, 14bitr3i 179 . . . . . . 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 287 . . . . . . 7  |-  ( ( ch  /\  ( ph  /\  ( x  =  A  /\  y  =  B ) ) )  ->  ps )
1815, 17sylbir 129 . . . . . 6  |-  ( ( ( ph  /\  ch )  /\  ( x  =  A  /\  y  =  B ) )  ->  ps )
1918ex 112 . . . . 5  |-  ( (
ph  /\  ch )  ->  ( ( x  =  A  /\  y  =  B )  ->  ps ) )
2011, 19eximd 1519 . . . 4  |-  ( (
ph  /\  ch )  ->  ( E. y ( x  =  A  /\  y  =  B )  ->  E. y ps )
)
218, 20eximd 1519 . . 3  |-  ( (
ph  /\  ch )  ->  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y ps ) )
2221impancom 251 . 2  |-  ( (
ph  /\  E. x E. y ( x  =  A  /\  y  =  B ) )  -> 
( ch  ->  E. x E. y ps ) )
235, 22sylan2 274 1  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( ch  ->  E. x E. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   F/wnf 1365   E.wex 1397    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  cnvoprab  5883
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