Theorem stcltr1i 29133

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

Step	Hyp	Ref	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	. . 3 |- A e. CH
3		stcltr1.3	. . 3 |- B e. CH
4		fveq2 6191	. . . . . . 7 |- ( x = A -> ( S ` x ) = ( S ` A ) )
5	4	eqeq1d 2624	. . . . . 6 |- ( x = A -> ( ( S ` x ) = 1 <-> ( S ` A ) = 1 ) )
6	5	imbi1d 331	. . . . 5 |- ( x = A -> ( ( ( S ` x ) = 1 -> ( S ` y ) = 1 ) <-> ( ( S ` A ) = 1 -> ( S ` y ) = 1 ) ) )
7		sseq1 3626	. . . . 5 |- ( x = A -> ( x C_ y <-> A C_ y ) )
8	6, 7	imbi12d 334	. . . 4 |- ( x = A -> ( ( ( ( S ` x ) = 1 -> ( S ` y ) = 1 ) -> x C_ y ) <-> ( ( ( S ` A ) = 1 -> ( S ` y ) = 1 ) -> A C_ y ) ) )
9		fveq2 6191	. . . . . . 7 |- ( y = B -> ( S ` y ) = ( S ` B ) )
10	9	eqeq1d 2624	. . . . . 6 |- ( y = B -> ( ( S ` y ) = 1 <-> ( S ` B ) = 1 ) )
11	10	imbi2d 330	. . . . 5 |- ( y = B -> ( ( ( S ` A ) = 1 -> ( S ` y ) = 1 ) <-> ( ( S ` A ) = 1 -> ( S ` B ) = 1 ) ) )
12		sseq2 3627	. . . . 5 |- ( y = B -> ( A C_ y <-> A C_ B ) )
13	11, 12	imbi12d 334	. . . 4 |- ( y = B -> ( ( ( ( S ` A ) = 1 -> ( S ` y ) = 1 ) -> A C_ y ) <-> ( ( ( S ` A ) = 1 -> ( S ` B ) = 1 ) -> A C_ B ) ) )
14	8, 13	rspc2v 3322	. . 3 |- ( ( 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 ) ) )
15	2, 3, 14	mp2an 708	. 2 |- ( 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 ) )
16	1, 15	simplbiim 659	1 |- ( ph -> ( ( ( S ` A ) = 1 -> ( S ` B ) = 1 ) -> A C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888   1c1 9937   CHcch 27786   Statescst 27819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  stcltr2i  29134  stcltrlem2  29136