Theorem imp 157

Description: Importation deduction. (Contributed by Mario Carneiro, 9-Oct-2014.)

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 55	. 2 |- (R, S) |= [S ==> T]
7	1, 5, 6	mpd 156	1 |- (R, S) |= T

Syntax hints:  *hb 3  [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:  con2d  161  exlimdv  167  alnex  186  notnot  200  ax3  205