Theorem mpd 146

Description: Modus ponens.

Hypotheses

Ref	Expression
mp.1	|- T:*
mp.2	|- R |= S
mp.3	|- R |= [S ==> T]

Assertion

Ref	Expression
mpd	|- R |= T

Proof of Theorem mpd

Step	Hyp	Ref	Expression
1		mp.2	. . . 4 |- R |= S
2		mp.3	. . . . 5 |- R |= [S ==> T]
3	1	ax-cb1 29	. . . . . 6 |- R:*
4	1	ax-cb2 30	. . . . . . 7 |- S:*
5		mp.1	. . . . . . 7 |- T:*
6	4, 5	imval 136	. . . . . 6 |- T. |= [[S ==> T] = [[S /\ T] = S]]
7	3, 6	a1i 28	. . . . 5 |- R |= [[S ==> T] = [[S /\ T] = S]]
8	2, 7	mpbi 72	. . . 4 |- R |= [[S /\ T] = S]
9	1, 8	mpbir 77	. . 3 |- R |= [S /\ T]
10	4, 5	dfan2 144	. . . 4 |- T. |= [[S /\ T] = (S, T)]
11	3, 10	a1i 28	. . 3 |- R |= [[S /\ T] = (S, T)]
12	9, 11	mpbi 72	. 2 |- R |= (S, T)
13	12	simprd 36	1 |- R |= T

Syntax hints:  *hb 3   = ke 7  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   /\ tan 109   ==> 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:  imp  147  notval2  149  notnot1  150  ecase  153  olc  154  orc  155  exlimdv2  156  ax4e  158  exlimd  171  ac  184  exmid  186  ax2  191  axmp  193  ax5  194  ax11  201