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Theorem nanbi 1294
Description: Show equivalence between the bidirectional and the Nicod version. (Contributed by Jeff Hoffman, 19-Nov-2007.)
Assertion
Ref Expression
nanbi ((φψ) ↔ ((φ ψ) ((φ φ) (ψ ψ))))

Proof of Theorem nanbi
StepHypRef Expression
1 pm4.57 483 . 2 (¬ (¬ (φ ψ) ¬ (¬ φ ¬ ψ)) ↔ ((φ ψ) φ ¬ ψ)))
2 df-nan 1288 . . 3 (((φ ψ) ((φ φ) (ψ ψ))) ↔ ¬ ((φ ψ) ((φ φ) (ψ ψ))))
3 df-nan 1288 . . . 4 ((φ ψ) ↔ ¬ (φ ψ))
4 df-nan 1288 . . . . 5 (((φ φ) (ψ ψ)) ↔ ¬ ((φ φ) (ψ ψ)))
5 nannot 1293 . . . . . 6 φ ↔ (φ φ))
6 nannot 1293 . . . . . 6 ψ ↔ (ψ ψ))
75, 6anbi12i 678 . . . . 5 ((¬ φ ¬ ψ) ↔ ((φ φ) (ψ ψ)))
84, 7xchbinxr 302 . . . 4 (((φ φ) (ψ ψ)) ↔ ¬ (¬ φ ¬ ψ))
93, 8anbi12i 678 . . 3 (((φ ψ) ((φ φ) (ψ ψ))) ↔ (¬ (φ ψ) ¬ (¬ φ ¬ ψ)))
102, 9xchbinx 301 . 2 (((φ ψ) ((φ φ) (ψ ψ))) ↔ ¬ (¬ (φ ψ) ¬ (¬ φ ¬ ψ)))
11 dfbi3 863 . 2 ((φψ) ↔ ((φ ψ) φ ¬ ψ)))
121, 10, 113bitr4ri 269 1 ((φψ) ↔ ((φ ψ) ((φ φ) (ψ ψ))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 176   wo 357   wa 358   wnan 1287
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 177  df-or 359  df-an 360  df-nan 1288
This theorem is referenced by:  nic-dfim  1434  nic-dfneg  1435
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