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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
StepHypRef Expression
1 imp.2 . 2 |- T:*
2 imp.3 . . . 4 |- R |= [S ==> T]
32ax-cb1 29 . . 3 |- R:*
4 imp.1 . . 3 |- S:*
53, 4simpr 23 . 2 |- (R, S) |= S
62, 4adantr 55 . 2 |- (R, S) |= [S ==> T]
71, 5, 6mpd 156 1 |- (R, S) |= T
Colors of variables: type var term
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
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