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Theorem stcltr1i 23777
Description: Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
stcltr1.1  |-  ( ph  <->  ( S  e.  States  /\  A. x  e.  CH  A. y  e.  CH  ( ( ( S `  x )  =  1  ->  ( S `  y )  =  1 )  ->  x  C_  y ) ) )
stcltr1.2  |-  A  e. 
CH
stcltr1.3  |-  B  e. 
CH
Assertion
Ref Expression
stcltr1i  |-  ( ph  ->  ( ( ( S `
 A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B
) )
Distinct variable groups:    x, y, A    x, B, y    x, S, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem stcltr1i
StepHypRef Expression
1 stcltr1.1 . 2  |-  ( ph  <->  ( S  e.  States  /\  A. x  e.  CH  A. y  e.  CH  ( ( ( S `  x )  =  1  ->  ( S `  y )  =  1 )  ->  x  C_  y ) ) )
2 stcltr1.2 . . . 4  |-  A  e. 
CH
3 stcltr1.3 . . . 4  |-  B  e. 
CH
4 fveq2 5728 . . . . . . . 8  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
54eqeq1d 2444 . . . . . . 7  |-  ( x  =  A  ->  (
( S `  x
)  =  1  <->  ( S `  A )  =  1 ) )
65imbi1d 309 . . . . . 6  |-  ( x  =  A  ->  (
( ( S `  x )  =  1  ->  ( S `  y )  =  1 )  <->  ( ( S `
 A )  =  1  ->  ( S `  y )  =  1 ) ) )
7 sseq1 3369 . . . . . 6  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
86, 7imbi12d 312 . . . . 5  |-  ( x  =  A  ->  (
( ( ( S `
 x )  =  1  ->  ( S `  y )  =  1 )  ->  x  C_  y
)  <->  ( ( ( S `  A )  =  1  ->  ( S `  y )  =  1 )  ->  A  C_  y ) ) )
9 fveq2 5728 . . . . . . . 8  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
109eqeq1d 2444 . . . . . . 7  |-  ( y  =  B  ->  (
( S `  y
)  =  1  <->  ( S `  B )  =  1 ) )
1110imbi2d 308 . . . . . 6  |-  ( y  =  B  ->  (
( ( S `  A )  =  1  ->  ( S `  y )  =  1 )  <->  ( ( S `
 A )  =  1  ->  ( S `  B )  =  1 ) ) )
12 sseq2 3370 . . . . . 6  |-  ( y  =  B  ->  ( A  C_  y  <->  A  C_  B
) )
1311, 12imbi12d 312 . . . . 5  |-  ( y  =  B  ->  (
( ( ( S `
 A )  =  1  ->  ( S `  y )  =  1 )  ->  A  C_  y
)  <->  ( ( ( S `  A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B ) ) )
148, 13rspc2v 3058 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A. x  e. 
CH  A. y  e.  CH  ( ( ( S `
 x )  =  1  ->  ( S `  y )  =  1 )  ->  x  C_  y
)  ->  ( (
( S `  A
)  =  1  -> 
( S `  B
)  =  1 )  ->  A  C_  B
) ) )
152, 3, 14mp2an 654 . . 3  |-  ( A. x  e.  CH  A. y  e.  CH  ( ( ( S `  x )  =  1  ->  ( S `  y )  =  1 )  ->  x  C_  y )  -> 
( ( ( S `
 A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B
) )
1615adantl 453 . 2  |-  ( ( S  e.  States  /\  A. x  e.  CH  A. y  e.  CH  ( ( ( S `  x )  =  1  ->  ( S `  y )  =  1 )  ->  x  C_  y ) )  ->  ( ( ( S `  A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B ) )
171, 16sylbi 188 1  |-  ( ph  ->  ( ( ( S `
 A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ` cfv 5454   1c1 8991   CHcch 22432   Statescst 22465
This theorem is referenced by:  stcltr2i  23778  stcltrlem2  23780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462
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