Theorem ax3 192

Description: Axiom Transp. Axiom A3 of [Margaris] p. 49.

Hypotheses

Ref	Expression
ax3.1	|- R:*
ax3.2	|- S:*

Assertion

Ref	Expression
ax3	|- T. |= [[(~ R) ==> (~ S)] ==> [S ==> R]]

Proof of Theorem ax3

Step	Hyp	Ref	Expression
1		ax3.1	. . . . 5 |- R:*
2		wnot 128	. . . . . 6 |- ~ :(* -> *)
3	2, 1	wc 45	. . . . 5 |- (~ R):*
4		wim 127	. . . . . . . 8 |- ==> :(* -> (* -> *))
5		ax3.2	. . . . . . . . 9 |- S:*
6	2, 5	wc 45	. . . . . . . 8 |- (~ S):*
7	4, 3, 6	wov 64	. . . . . . 7 |- [(~ R) ==> (~ S)]:*
8	7, 5	wct 44	. . . . . 6 |- ([(~ R) ==> (~ S)], S):*
9	1	exmid 186	. . . . . 6 |- T. |= [R \/ (~ R)]
10	8, 9	a1i 28	. . . . 5 |- ([(~ R) ==> (~ S)], S) |= [R \/ (~ R)]
11	10	ax-cb1 29	. . . . . 6 |- ([(~ R) ==> (~ S)], S):*
12	11, 1	simpr 23	. . . . 5 |- (([(~ R) ==> (~ S)], S), R) |= R
13		wfal 125	. . . . . . . 8 |- F.:*
14	7	id 25	. . . . . . . . . 10 |- [(~ R) ==> (~ S)] |= [(~ R) ==> (~ S)]
15	3, 6, 14	imp 147	. . . . . . . . 9 |- ([(~ R) ==> (~ S)], (~ R)) |= (~ S)
16	15	ax-cb1 29	. . . . . . . . . 10 |- ([(~ R) ==> (~ S)], (~ R)):*
17	5	notval 135	. . . . . . . . . 10 |- T. |= [(~ S) = [S ==> F.]]
18	16, 17	a1i 28	. . . . . . . . 9 |- ([(~ R) ==> (~ S)], (~ R)) |= [(~ S) = [S ==> F.]]
19	15, 18	mpbi 72	. . . . . . . 8 |- ([(~ R) ==> (~ S)], (~ R)) |= [S ==> F.]
20	5, 13, 19	imp 147	. . . . . . 7 |- (([(~ R) ==> (~ S)], (~ R)), S) |= F.
21	20	an32s 55	. . . . . 6 |- (([(~ R) ==> (~ S)], S), (~ R)) |= F.
22	1	pm2.21 143	. . . . . 6 |- F. |= R
23	21, 22	syl 16	. . . . 5 |- (([(~ R) ==> (~ S)], S), (~ R)) |= R
24	1, 3, 1, 10, 12, 23	ecase 153	. . . 4 |- ([(~ R) ==> (~ S)], S) |= R
25	24	ex 148	. . 3 |- [(~ R) ==> (~ S)] |= [S ==> R]
26		wtru 40	. . 3 |- T.:*
27	25, 26	adantl 51	. 2 |- (T., [(~ R) ==> (~ S)]) |= [S ==> R]
28	27	ex 148	1 |- T. |= [[(~ R) ==> (~ S)] ==> [S ==> R]]

Syntax hints:  *hb 3  kc 5   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  ~ tne 110   ==> tim 111   \/ tor 114

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-ac 183

This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120  df-or 122

This theorem is referenced by: (None)