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

Theorem mpbidi 231
 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 219 . 2 (𝜑 → (𝜓𝜒))
41, 3sylcom 30 1 (𝜃 → (𝜑𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196 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 197 This theorem is referenced by:  ralxfr2d  4912  ovmpt4g  6825  ov3  6839  omeulem2  7708  domtriomlem  9302  nsmallnq  9837  bposlem1  25054  pntrsumbnd  25300  mptsnunlem  33315  poimirlem27  33566  frege92  38566  nzss  38833  setis  42769
 Copyright terms: Public domain W3C validator