Theorem con3ALTVD 45836

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 154). 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 45451 is con3ALTVD 45836 without virtual deductions and was automatically derived from con3ALTVD 45836. 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 45495	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ▶   (𝜑 → 𝜓)   )
2		idn2 45534	. . . . . . 7 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜑   )
3		notnotr 131	. . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑)
4	2, 3	e2 45552	. . . . . 6 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜑   )
5		id 23	. . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
6	1, 4, 5	e12 45644	. . . . 5 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶   𝜓   )
7		notnot 143	. . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓)
8	6, 7	e2 45552	. . . 4 ⊢ (   (𝜑 → 𝜓)   ,    ¬ ¬ 𝜑   ▶    ¬ ¬ 𝜓   )
9	8	in2 45526	. . 3 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ ¬ 𝜑 → ¬ ¬ 𝜓)   )
10		con4 114	. . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
11	9, 10	e1a 45548	. 2 ⊢ (   (𝜑 → 𝜓)   ▶   (¬ 𝜓 → ¬ 𝜑)   )
12	11	in1 45492	1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-vd1 45491  df-vd2 45499

This theorem is used by: (None)