LLPE Home Linear Logic Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  LLPE Home  >  Th. List  >  ilbd Structured version  

Theorem ilbd 97
Description: Construct a biconditional from its forward and reverse implications.
Hypotheses
Ref Expression
ilbd.1 (𝜑 ⊸ (𝜓𝜒))
ilbd.2 (𝜑 ⊸ (𝜒𝜓))
Assertion
Ref Expression
ilbd (𝜑 ⊸ (𝜓𝜒))

Proof of Theorem ilbd
StepHypRef Expression
1 ilbd.1 . . . . 5 (𝜑 ⊸ (𝜓𝜒))
21dfli1i 65 . . . 4 (~ 𝜑 ⅋ (𝜓𝜒))
3 ilbd.2 . . . . 5 (𝜑 ⊸ (𝜒𝜓))
43dfli1i 65 . . . 4 (~ 𝜑 ⅋ (𝜒𝜓))
52, 4ax-iac 32 . . 3 (~ 𝜑 ⅋ ((𝜓𝜒) & (𝜒𝜓)))
6 dflb 93 . . 3 ((𝜓𝜒) ⧟ ((𝜓𝜒) & (𝜒𝜓)))
75, 6lb2d 58 . 2 (~ 𝜑 ⅋ (𝜓𝜒))
87dfli2i 66 1 (𝜑 ⊸ (𝜓𝜒))
Colors of variables: wff var nilad
Syntax hints:  ~ wneg 3   & wac 30  wlb 55  wli 61
This theorem was proved from axioms:  ax-ibot 4  ax-ebot 5  ax-cut 6  ax-init 7  ax-mdco 8  ax-mdas 9  ax-iac 32  ax-eac1 33  ax-eac2 34
This theorem depends on definitions:  df-lb 56  df-li 62
This theorem is referenced by:  lbsymd  102
  Copyright terms: Public domain W3C validator