Theorem ax2 204

Description: Axiom Frege. Axiom A2 of [Margaris] p. 49. (Contributed by Mario Carneiro, 9-Oct-2014.)

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 137	. . . . . . . . . 10 |- ==> :(* -> (* -> *))
4		ax1.1	. . . . . . . . . 10 |- R:*
5	3, 2, 1	wov 72	. . . . . . . . . 10 |- [S ==> T]:*
6	3, 4, 5	wov 72	. . . . . . . . 9 |- [R ==> [S ==> T]]:*
7	3, 4, 2	wov 72	. . . . . . . . 9 |- [R ==> S]:*
8	6, 7	wct 48	. . . . . . . 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 38	. . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [R ==> S]
12	2, 9, 11	mpd 156	. . . . . 6 |- (([R ==> [S ==> T]], [R ==> S]), R) |= S
13	10	simpld 37	. . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [R ==> [S ==> T]]
14	5, 9, 13	mpd 156	. . . . . 6 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [S ==> T]
15	1, 12, 14	mpd 156	. . . . 5 |- (([R ==> [S ==> T]], [R ==> S]), R) |= T
16	15	ex 158	. . . 4 |- ([R ==> [S ==> T]], [R ==> S]) |= [R ==> T]
17	16	ex 158	. . 3 |- [R ==> [S ==> T]] |= [[R ==> S] ==> [R ==> T]]
18		wtru 43	. . 3 |- T.:*
19	17, 18	adantl 56	. 2 |- (T., [R ==> [S ==> T]]) |= [[R ==> S] ==> [R ==> T]]
20	19	ex 158	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 121

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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-an 128  df-im 129

This theorem is referenced by: (None)