MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dedlem0a Structured version   Visualization version   GIF version

Theorem dedlem0a 1039
Description: Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
Assertion
Ref Expression
dedlem0a (𝜑 → (𝜓 ↔ ((𝜒𝜑) → (𝜓𝜑))))

Proof of Theorem dedlem0a
StepHypRef Expression
1 iba 531 . 2 (𝜑 → (𝜓 ↔ (𝜓𝜑)))
2 biimt 364 . . 3 ((𝜒𝜑) → ((𝜓𝜑) ↔ ((𝜒𝜑) → (𝜓𝜑))))
32jarri 107 . 2 (𝜑 → ((𝜓𝜑) ↔ ((𝜒𝜑) → (𝜓𝜑))))
41, 3bitrd 282 1 (𝜑 → (𝜓 ↔ ((𝜒𝜑) → (𝜓𝜑))))
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:  iftrue  4434
  Copyright terms: Public domain W3C validator