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Theorem con3ALTVD 39651
Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 7 of Section 14 of [Margaris] p. 60 ( which is con3 149). The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con3ALT2 39238 is con3ALTVD 39651 without virtual deductions and was automatically derived from con3ALTVD 39651. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
 1:: ⊢ (   (𝜑 → 𝜓)   ▶   (𝜑 → 𝜓)   ) 2:: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜑   ) 3:: ⊢ (¬ ¬ 𝜑 → 𝜑) 4:2: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜑   ) 5:1,4: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜓   ) 6:: ⊢ (𝜓 → ¬ ¬ 𝜓) 7:6,5: ⊢ (   (𝜑 → 𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜓   ) 8:7: ⊢ (   (𝜑 → 𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓 )   ) 9:: ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑)) 10:8: ⊢ (   (𝜑 → 𝜓)   ▶   (¬ 𝜓 → ¬ 𝜑)   ) qed:10: ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
con3ALTVD ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3ALTVD
StepHypRef Expression
1 idn1 39292 . . . . . 6 (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2 idn2 39340 . . . . . . 7 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
3 notnotr 125 . . . . . . 7 (¬ ¬ 𝜑𝜑)
42, 3e2 39358 . . . . . 6 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜑   )
5 id 22 . . . . . 6 ((𝜑𝜓) → (𝜑𝜓))
61, 4, 5e12 39453 . . . . 5 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜓   )
7 notnot 136 . . . . 5 (𝜓 → ¬ ¬ 𝜓)
86, 7e2 39358 . . . 4 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜓   )
98in2 39332 . . 3 (   (𝜑𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓)   )
10 con4 112 . . 3 ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
119, 10e1a 39354 . 2 (   (𝜑𝜓)   ▶   𝜓 → ¬ 𝜑)   )
1211in1 39289 1 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 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 197  df-an 385  df-vd1 39288  df-vd2 39296 This theorem is referenced by: (None)
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