Theorem ax2 191

Description: Axiom Frege. Axiom A2 of [Margaris] p. 49.

Hypotheses

Ref	Expression
ax1.1	|- R:*
ax1.2	|- S:*
ax2.3	|- T:*

Assertion

Ref	Expression
ax2	|- T. |= [[R ==> [S ==> T]] ==> [[R ==> S] ==> [R ==> T]]]

Proof of Theorem ax2

Step	Hyp	Ref	Expression
1		ax2.3	. . . . . 6 |- T:*
2		ax1.2	. . . . . . 7 |- S:*
3		wim 127	. . . . . . . . . 10 |- ==> :(* -> (* -> *))
4		ax1.1	. . . . . . . . . 10 |- R:*
5	3, 2, 1	wov 64	. . . . . . . . . 10 |- [S ==> T]:*
6	3, 4, 5	wov 64	. . . . . . . . 9 |- [R ==> [S ==> T]]:*
7	3, 4, 2	wov 64	. . . . . . . . 9 |- [R ==> S]:*
8	6, 7	wct 44	. . . . . . . 8 |- ([R ==> [S ==> T]], [R ==> S]):*
9	8, 4	simpr 23	. . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= R
10	8, 4	simpl 22	. . . . . . . 8 |- (([R ==> [S ==> T]], [R ==> S]), R) |= ([R ==> [S ==> T]], [R ==> S])
11	10	simprd 36	. . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [R ==> S]
12	2, 9, 11	mpd 146	. . . . . 6 |- (([R ==> [S ==> T]], [R ==> S]), R) |= S
13	10	simpld 35	. . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [R ==> [S ==> T]]
14	5, 9, 13	mpd 146	. . . . . 6 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [S ==> T]
15	1, 12, 14	mpd 146	. . . . 5 |- (([R ==> [S ==> T]], [R ==> S]), R) |= T
16	15	ex 148	. . . 4 |- ([R ==> [S ==> T]], [R ==> S]) |= [R ==> T]
17	16	ex 148	. . 3 |- [R ==> [S ==> T]] |= [[R ==> S] ==> [R ==> T]]
18		wtru 40	. . 3 |- T.:*
19	17, 18	adantl 51	. 2 |- (T., [R ==> [S ==> T]]) |= [[R ==> S] ==> [R ==> T]]
20	19	ex 148	1 |- T. |= [[R ==> [S ==> T]] ==> [[R ==> S] ==> [R ==> T]]]

Syntax hints:  *hb 3  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 111

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

This theorem depends on definitions:  df-ov 65  df-an 118  df-im 119

This theorem is referenced by: (None)