Theorem imp 147

Description: Importation deduction.

Hypotheses

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

Assertion

Ref	Expression
imp	|- (R, S) |= T

Proof of Theorem imp

Step	Hyp	Ref	Expression
1		imp.2	. 2 |- T:*
2		imp.3	. . . 4 |- R |= [S ==> T]
3	2	ax-cb1 29	. . 3 |- R:*
4		imp.1	. . 3 |- S:*
5	3, 4	simpr 23	. 2 |- (R, S) |= S
6	2, 4	adantr 50	. 2 |- (R, S) |= [S ==> T]
7	1, 5, 6	mpd 146	1 |- (R, S) |= T

Syntax hints:  *hb 3  [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:  con2d  151  exlimdv  157  alnex  174  notnot  187  ax3  192