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Theorem imnand2 31569
 Description: An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
Assertion
Ref Expression
imnand2 ((¬ 𝜑𝜓) ↔ ((𝜑𝜑) ⊼ (𝜓𝜓)))

Proof of Theorem imnand2
StepHypRef Expression
1 nannot 1445 . . . 4 𝜑 ↔ (𝜑𝜑))
2 nannot 1445 . . . 4 𝜓 ↔ (𝜓𝜓))
31, 2anbi12i 729 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ((𝜑𝜑) ∧ (𝜓𝜓)))
43notbii 309 . 2 (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ ((𝜑𝜑) ∧ (𝜓𝜓)))
5 iman 439 . 2 ((¬ 𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))
6 df-nan 1440 . 2 (((𝜑𝜑) ⊼ (𝜓𝜓)) ↔ ¬ ((𝜑𝜑) ∧ (𝜓𝜓)))
74, 5, 63bitr4i 291 1 ((¬ 𝜑𝜓) ↔ ((𝜑𝜑) ⊼ (𝜓𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ⊼ wnan 1439 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 196  df-an 385  df-nan 1440 This theorem is referenced by: (None)
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