Theorem stcltr1 10201

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 groups:   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 3724	. . . . . . . 8 |- (x = A -> (S` x) = (S` A))
5	4	eqeq1d 1483	. . . . . . 7 |- (x = A -> ((S` x) = 1 <-> (S` A) = 1))
6	5	imbi1d 613	. . . . . 6 |- (x = A -> (((S` x) = 1 -> (S` y) = 1) <-> ((S` A) = 1 -> (S` y) = 1)))
7		sseq1 2082	. . . . . 6 |- (x = A -> (x (_ y <-> A (_ y))
8	6, 7	imbi12d 626	. . . . 5 |- (x = A -> ((((S` x) = 1 -> (S` y) = 1) -> x (_ y) <-> (((S` A) = 1 -> (S` y) = 1) -> A (_ y)))
9		fveq2 3724	. . . . . . . 8 |- (y = B -> (S` y) = (S` B))
10	9	eqeq1d 1483	. . . . . . 7 |- (y = B -> ((S` y) = 1 <-> (S` B) = 1))
11	10	imbi2d 612	. . . . . 6 |- (y = B -> (((S` A) = 1 -> (S` y) = 1) <-> ((S` A) = 1 -> (S` B) = 1)))
12		sseq2 2083	. . . . . 6 |- (y = B -> (A (_ y <-> A (_ B))
13	11, 12	imbi12d 626	. . . . 5 |- (y = B -> ((((S` A) = 1 -> (S` y) = 1) -> A (_ y) <-> (((S` A) = 1 -> (S` B) = 1) -> A (_ B)))
14	8, 13	rcla42v 1880	. . . 4 |- ((A e. CH /\ B e. CH) -> (A.x e. CH A.y e. CH (((S` x) = 1 -> (S` y) = 1) -> x (_ y) -> (((S` A) = 1 -> (S` B) = 1) -> A (_ B)))
15	2, 3, 14	mp2an 697	. . 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 388	. 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 199	1 |- (ph -> (((S` A) = 1 -> (S` B) = 1) -> A (_ B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  ` cfv 3182  1c1 5235  CHcch 8798  Statescst 8831

This theorem is referenced by:  stcltr2 10202  stcltrlem2 10204

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198

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