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

Theorem nanbi 1484
Description: Biconditional in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
Assertion
Ref Expression
nanbi ((𝜑𝜓) ↔ ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))))

Proof of Theorem nanbi
StepHypRef Expression
1 dfbi3 1041 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 df-or 842 . . 3 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ (¬ (𝜑𝜓) → (¬ 𝜑 ∧ ¬ 𝜓)))
3 df-nan 1476 . . . . 5 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
43bicomi 225 . . . 4 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
5 nannot 1483 . . . . 5 𝜑 ↔ (𝜑𝜑))
6 nannot 1483 . . . . 5 𝜓 ↔ (𝜓𝜓))
75, 6anbi12i 626 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ((𝜑𝜑) ∧ (𝜓𝜓)))
84, 7imbi12i 352 . . 3 ((¬ (𝜑𝜓) → (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑𝜓) → ((𝜑𝜑) ∧ (𝜓𝜓))))
91, 2, 83bitri 298 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) → ((𝜑𝜑) ∧ (𝜓𝜓))))
10 nannan 1481 . 2 (((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))) ↔ ((𝜑𝜓) → ((𝜑𝜑) ∧ (𝜓𝜓))))
119, 10bitr4i 279 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  wnan 1475
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 208  df-an 397  df-or 842  df-nan 1476
This theorem is referenced by:  nic-dfim  1661  nic-dfneg  1662
  Copyright terms: Public domain W3C validator