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Theorem spc2d 28105
Description: 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
spc2d  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( A. x A. y ps  ->  ch )
)
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 spc2d
StepHypRef Expression
1 2nalexn 1694 . . 3  |-  ( -. 
A. x A. y ps 
<->  E. x E. y  -.  ps )
21con1bii 332 . 2  |-  ( -. 
E. x E. y  -.  ps  <->  A. x A. y ps )
3 spc2ed.x . . . . 5  |-  F/ x ch
43nfn 1960 . . . 4  |-  F/ x  -.  ch
5 spc2ed.y . . . . 5  |-  F/ y ch
65nfn 1960 . . . 4  |-  F/ y  -.  ch
7 spc2ed.1 . . . . 5  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ps  <->  ch )
)
87notbid 295 . . . 4  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( -.  ps  <->  -.  ch )
)
94, 6, 8spc2ed 28104 . . 3  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( -.  ch  ->  E. x E. y  -. 
ps ) )
109con1d 127 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( -.  E. x E. y  -.  ps  ->  ch ) )
112, 10syl5bir 221 1  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( A. x A. y ps  ->  ch )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657   F/wnf 1661    e. wcel 1872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3082
This theorem is referenced by: (None)
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