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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⊧[ST]
Assertion
Ref Expression
imp (R, S)⊧T

Proof of Theorem imp
StepHypRef Expression
1 imp.2 . 2 T:∗
2 imp.3 . . . 4 R⊧[ST]
32ax-cb1 29 . . 3 R:∗
4 imp.1 . . 3 S:∗
53, 4simpr 23 . 2 (R, S)⊧S
62, 4adantr 55 . 2 (R, S)⊧[ST]
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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