HOLE Home Higher-Order Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HOLE Home  >  Th. List  >  mpdan Unicode version

Theorem mpdan 35
Description: Modus ponens deduction. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
mpdan.1 |- R |= S
mpdan.2 |- (R, S) |= T
Assertion
Ref Expression
mpdan |- R |= T

Proof of Theorem mpdan
StepHypRef Expression
1 mpdan.1 . . . 4 |- R |= S
21ax-cb1 29 . . 3 |- R:*
32id 25 . 2 |- R |= R
4 mpdan.2 . 2 |- (R, S) |= T
53, 1, 4syl2anc 19 1 |- R |= T
Colors of variables: type var term
Syntax hints:  kct 10   |= wffMMJ2 11
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-id 24  ax-cb1 29
This theorem is referenced by:  hbov  111  hbct  155
  Copyright terms: Public domain W3C validator