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Theorem stcltr1i 22854
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 5525 . . . . . . . 8  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
54eqeq1d 2291 . . . . . . 7  |-  ( x  =  A  ->  (
( S `  x
)  =  1  <->  ( S `  A )  =  1 ) )
65imbi1d 308 . . . . . 6  |-  ( x  =  A  ->  (
( ( S `  x )  =  1  ->  ( S `  y )  =  1 )  <->  ( ( S `
 A )  =  1  ->  ( S `  y )  =  1 ) ) )
7 sseq1 3199 . . . . . 6  |-  ( x  =  A  ->  (
x  C_  y  <->  A  C_  y
) )
86, 7imbi12d 311 . . . . 5  |-  ( x  =  A  ->  (
( ( ( S `
 x )  =  1  ->  ( S `  y )  =  1 )  ->  x  C_  y
)  <->  ( ( ( S `  A )  =  1  ->  ( S `  y )  =  1 )  ->  A  C_  y ) ) )
9 fveq2 5525 . . . . . . . 8  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
109eqeq1d 2291 . . . . . . 7  |-  ( y  =  B  ->  (
( S `  y
)  =  1  <->  ( S `  B )  =  1 ) )
1110imbi2d 307 . . . . . 6  |-  ( y  =  B  ->  (
( ( S `  A )  =  1  ->  ( S `  y )  =  1 )  <->  ( ( S `
 A )  =  1  ->  ( S `  B )  =  1 ) ) )
12 sseq2 3200 . . . . . 6  |-  ( y  =  B  ->  ( A  C_  y  <->  A  C_  B
) )
1311, 12imbi12d 311 . . . . 5  |-  ( y  =  B  ->  (
( ( ( S `
 A )  =  1  ->  ( S `  y )  =  1 )  ->  A  C_  y
)  <->  ( ( ( S `  A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B ) ) )
148, 13rspc2v 2890 . . . 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 653 . . 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 452 . 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 187 1  |-  ( ph  ->  ( ( ( S `
 A )  =  1  ->  ( S `  B )  =  1 )  ->  A  C_  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255   1c1 8738   CHcch 21509   Statescst 21542
This theorem is referenced by:  stcltr2i  22855  stcltrlem2  22857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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