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

Theorem mpbiran2d 707
 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 706 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  5830  funsnfsupp  8843  discld  21701  cncffvrn  23510  itgfsum  24437  dchreq  25849  lgsneg  25912  lgsquadlem2  25972  dfconngr1  27980  cover2  35168
 Copyright terms: Public domain W3C validator