Theorem stcltr1 5707

Description: Property of a strong classical state.

Hypotheses

Ref	Expression
stcltr1.1	|- (ph <-> (S e. States /\ A.x e. CH A.y e. CH (((S` x) = 1 -> (S` y) = 1) -> x (_ y)))
stcltr1.2	|- A e. CH
stcltr1.3	|- B e. CH

Assertion

Ref	Expression
stcltr1	|- (ph -> (((S` A) = 1 -> (S` B) = 1) -> A (_ B))

Distinct variable group(s):   x,y,A   x,B,y   x,S,y

Proof of Theorem stcltr1

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 (_ y)))
2		stcltr1.2	. . . 4 |- A e. CH
3		stcltr1.3	. . . 4 |- B e. CH
4		fveq2 2832	. . . . . . . 8 |- (x = A -> (S` x) = (S` A))
5	4	cleq1d 1109	. . . . . . 7 |- (x = A -> ((S` x) = 1 <-> (S` A) = 1))
6	5	imbi1d 465	. . . . . 6 |- (x = A -> (((S` x) = 1 -> (S` y) = 1) <-> ((S` A) = 1 -> (S` y) = 1)))
7		sseq1 1521	. . . . . 6 |- (x = A -> (x (_ y <-> A (_ y))
8	6, 7	imbi12d 474	. . . . 5 |- (x = A -> ((((S` x) = 1 -> (S` y) = 1) -> x (_ y) <-> (((S` A) = 1 -> (S` y) = 1) -> A (_ y)))
9		fveq2 2832	. . . . . . . 8 |- (y = B -> (S` y) = (S` B))
10	9	cleq1d 1109	. . . . . . 7 |- (y = B -> ((S` y) = 1 <-> (S` B) = 1))
11	10	imbi2d 464	. . . . . 6 |- (y = B -> (((S` A) = 1 -> (S` y) = 1) <-> ((S` A) = 1 -> (S` B) = 1)))
12		sseq2 1522	. . . . . 6 |- (y = B -> (A (_ y <-> A (_ B))
13	11, 12	imbi12d 474	. . . . 5 |- (y = B -> ((((S` A) = 1 -> (S` y) = 1) -> A (_ y) <-> (((S` A) = 1 -> (S` B) = 1) -> A (_ B)))
14	8, 13	rcla42v 1404	. . . 4 |- (A.x e. CH A.y e. CH (((S` x) = 1 -> (S` y) = 1) -> x (_ y) -> ((A e. CH /\ B e. CH) -> (((S` A) = 1 -> (S` B) = 1) -> A (_ B)))
15	2, 3, 14	mp2ani 523	. . 3 |- (A.x e. CH A.y e. CH (((S` x) = 1 -> (S` y) = 1) -> x (_ y) -> (((S` A) = 1 -> (S` B) = 1) -> A (_ B))
16	15	adantl 305	. 2 |- ((S e. States /\ A.x e. CH A.y e. CH (((S` x) = 1 -> (S` y) = 1) -> x (_ y)) -> (((S` A) = 1 -> (S` B) = 1) -> A (_ B))
17	1, 16	sylbi 174	1 |- (ph -> (((S` A) = 1 -> (S` B) = 1) -> A (_ B))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487  ` cfv 2422  1c1 4029  CHcch 4968  Statescst 4979

This theorem is referenced by:  stcltr2 5708  stcltrlem2 5710

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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