Theorem mpd 156

Description: Modus ponens. (Contributed by Mario Carneiro, 9-Oct-2014.)

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 146	. . . . . 6 |- T. |= [[S ==> T] = [[S /\ T] = S]]
7	3, 6	a1i 28	. . . . 5 |- R |= [[S ==> T] = [[S /\ T] = S]]
8	2, 7	mpbi 82	. . . 4 |- R |= [[S /\ T] = S]
9	1, 8	mpbir 87	. . 3 |- R |= [S /\ T]
10	4, 5	dfan2 154	. . . 4 |- T. |= [[S /\ T] = (S, T)]
11	3, 10	a1i 28	. . 3 |- R |= [[S /\ T] = (S, T)]
12	9, 11	mpbi 82	. 2 |- R |= (S, T)
13	12	simprd 38	1 |- R |= T

Syntax hints:  *hb 3   = ke 7  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   /\ tan 119   ==> 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:  imp  157  notval2  159  notnot1  160  ecase  163  olc  164  orc  165  exlimdv2  166  ax4e  168  exlimd  183  ac  197  exmid  199  ax2  204  axmp  206  ax5  207  ax11  214