Theorem con3ALTVD 45300

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 44915 is con3ALTVD 45300 without virtual deductions and was automatically derived from con3ALTVD 45300. 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 44959	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ▶   (𝜑 → 𝜓)   )
2		idn2 44998	. . . . . . 7 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
3		notnotr 130	. . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑)
4	2, 3	e2 45016	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜑   )
5		id 22	. . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
6	1, 4, 5	e12 45108	. . . . 5 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜓   )
7		notnot 142	. . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓)
8	6, 7	e2 45016	. . . 4 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜓   )
9	8	in2 44990	. . 3 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓)   )
10		con4 113	. . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
11	9, 10	e1a 45012	. 2 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ 𝜓 → ¬ 𝜑)   )
12	11	in1 44956	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 207  df-an 396  df-vd1 44955  df-vd2 44963

This theorem is referenced by: (None)