MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpbiran2d Structured version   Visualization version   GIF version

Theorem mpbiran2d 708
Description: Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.)
Hypotheses
Ref Expression
mpbiran2d.1 (𝜑𝜃)
mpbiran2d.2 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
Assertion
Ref Expression
mpbiran2d (𝜑 → (𝜓𝜒))

Proof of Theorem mpbiran2d
StepHypRef Expression
1 mpbiran2d.1 . 2 (𝜑𝜃)
2 mpbiran2d.2 . . 3 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
32biancomd 467 . 2 (𝜑 → (𝜓 ↔ (𝜃𝜒)))
41, 3mpbirand 707 1 (𝜑 → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399
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 210  df-an 400
This theorem is referenced by:  opelidres  5863  funsnfsupp  9009  discld  21986  cncffvrn  23795  itgfsum  24724  dchreq  26139  lgsneg  26202  lgsquadlem2  26262  dfconngr1  28271  cover2  35609  iscnrm3rlem6  45912  thincmon  45988  thincepi  45989
  Copyright terms: Public domain W3C validator