 Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bian1d Structured version   Visualization version   GIF version

Theorem bian1d 28478
 Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)
Hypothesis
Ref Expression
bian1d.1 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
Assertion
Ref Expression
bian1d (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))

Proof of Theorem bian1d
StepHypRef Expression
1 bian1d.1 . . . 4 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
21biimpd 217 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
32adantld 481 . 2 (𝜑 → ((𝜒𝜓) → (𝜒𝜃)))
4 simpl 471 . . . 4 ((𝜒𝜃) → 𝜒)
54a1i 11 . . 3 (𝜑 → ((𝜒𝜃) → 𝜒))
61biimprd 236 . . 3 (𝜑 → ((𝜒𝜃) → 𝜓))
75, 6jcad 553 . 2 (𝜑 → ((𝜒𝜃) → (𝜒𝜓)))
83, 7impbid 200 1 (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 194   ∧ wa 382 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  df-an 384 This theorem is referenced by:  funcnvmptOLD  28638  funcnvmpt  28639
 Copyright terms: Public domain W3C validator