Theorem con3ALTVD 43979

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 43593 is con3ALTVD 43979 without virtual deductions and was automatically derived from con3ALTVD 43979. 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 43637	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ▶   (𝜑 → 𝜓)   )
2		idn2 43676	. . . . . . 7 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
3		notnotr 130	. . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑)
4	2, 3	e2 43694	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜑   )
5		id 22	. . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
6	1, 4, 5	e12 43787	. . . . 5 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜓   )
7		notnot 142	. . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓)
8	6, 7	e2 43694	. . . 4 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜓   )
9	8	in2 43668	. . 3 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓)   )
10		con4 113	. . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
11	9, 10	e1a 43690	. 2 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ 𝜓 → ¬ 𝜑)   )
12	11	in1 43634	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 395  df-vd1 43633  df-vd2 43641

This theorem is referenced by: (None)