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Theorem con3ALTVD 37973
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 147). 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 37556 is con3ALTVD 37973 without virtual deductions and was automatically derived from con3ALTVD 37973. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
1:: (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2:: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜑   )
3:: (¬ ¬ 𝜑𝜑)
4:2: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜑   )
5:1,4: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   𝜓   )
6:: (𝜓 → ¬ ¬ 𝜓)
7:6,5: (   (𝜑𝜓)   ,   ¬ ¬ 𝜑   ▶   ¬ ¬ 𝜓   )
8:7: (   (𝜑𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓 )   )
9:: ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 ¬ 𝜑))
10:8: (   (𝜑𝜓)   ▶   𝜓 → ¬ 𝜑)   )
qed:10: ((𝜑𝜓) → (¬ 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 37610 . . . . . 6 (   (𝜑𝜓)   ▶   (𝜑𝜓)   )
2 idn2 37658 . . . . . . 7 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
3 notnotr 123 . . . . . . 7 (¬ ¬ 𝜑𝜑)
42, 3e2 37676 . . . . . 6 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜑   )
5 id 22 . . . . . 6 ((𝜑𝜓) → (𝜑𝜓))
61, 4, 5e12 37771 . . . . 5 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜓   )
7 notnot 134 . . . . 5 (𝜓 → ¬ ¬ 𝜓)
86, 7e2 37676 . . . 4 (   (𝜑𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜓   )
98in2 37650 . . 3 (   (𝜑𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓)   )
10 con4 110 . . 3 ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
119, 10e1a 37672 . 2 (   (𝜑𝜓)   ▶   𝜓 → ¬ 𝜑)   )
1211in1 37607 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 195  df-an 384  df-vd1 37606  df-vd2 37614
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator