Theorem con3ALTVD 45496

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 45111 is con3ALTVD 45496 without virtual deductions and was automatically derived from con3ALTVD 45496. 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 45155	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ▶   (𝜑 → 𝜓)   )
2		idn2 45194	. . . . . . 7 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
3		notnotr 130	. . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑)
4	2, 3	e2 45212	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜑   )
5		id 22	. . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
6	1, 4, 5	e12 45304	. . . . 5 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜓   )
7		notnot 142	. . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓)
8	6, 7	e2 45212	. . . 4 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜓   )
9	8	in2 45186	. . 3 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓)   )
10		con4 113	. . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
11	9, 10	e1a 45208	. 2 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ 𝜓 → ¬ 𝜑)   )
12	11	in1 45152	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 209  df-an 400  df-vd1 45151  df-vd2 45159

This theorem is referenced by: (None)