Theorem con3ALTVD 43667

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 153). 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 43281 is con3ALTVD 43667 without virtual deductions and was automatically derived from con3ALTVD 43667. 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	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3ALTVD

Step	Hyp	Ref	Expression
1		idn1 43325	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ▶   (𝜑 → 𝜓)   )
2		idn2 43364	. . . . . . 7 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
3		notnotr 130	. . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑)
4	2, 3	e2 43382	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜑   )
5		id 22	. . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
6	1, 4, 5	e12 43475	. . . . 5 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜓   )
7		notnot 142	. . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓)
8	6, 7	e2 43382	. . . 4 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜓   )
9	8	in2 43356	. . 3 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓)   )
10		con4 113	. . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
11	9, 10	e1a 43378	. 2 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ 𝜓 → ¬ 𝜑)   )
12	11	in1 43322	1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

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 206  df-an 397  df-vd1 43321  df-vd2 43329

This theorem is referenced by: (None)