MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpbidi Structured version   Visualization version   GIF version

Theorem mpbidi 229
Description: A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
Hypotheses
Ref Expression
mpbidi.min (𝜃 → (𝜑𝜓))
mpbidi.maj (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
mpbidi (𝜃 → (𝜑𝜒))

Proof of Theorem mpbidi
StepHypRef Expression
1 mpbidi.min . 2 (𝜃 → (𝜑𝜓))
2 mpbidi.maj . . 3 (𝜑 → (𝜓𝜒))
32biimpd 217 . 2 (𝜑 → (𝜓𝜒))
41, 3sylcom 30 1 (𝜃 → (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 195
This theorem is referenced by:  tpid3gOLD  4245  ralxfr2d  4800  ovmpt4g  6656  ov3  6670  omeulem2  7524  domtriomlem  9121  nsmallnq  9652  bposlem1  24723  pntrsumbnd  24969  mptsnunlem  32161  poimirlem27  32406  frege92  37069  nzss  37338
  Copyright terms: Public domain W3C validator